View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 32 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 13 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 6 :

*Example: Identify percent rate of change in functions such as *y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}*, and classify them as representing exponential growth or decay.* [MA2019] AL1-19 (9-12) 9 :

32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

Example: Identify percent rate of change in functions such as *y* = (1.02)^{t}, *y* = (0.97)^{t}, *y* = (1.01)^{12t}, and *y* = (1.2)^{t/10}, and classify them as representing exponential growth and decay.

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

In this video lesson, students begin to rewrite quadratic expressions from standard to factored form.

Students relate the numbers in the factored form to the coefficients of the terms in standard form, looking for a structure that can be used to go in reverse—from standard form to factored form (MP7).

(This lesson only looks at expressions of the form (*x* + *m*)(*x* + *n*) and (*x* – *m*)(*x* – *n*) where *m* and *n* are positive.)

32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

Example: Identify percent rate of change in functions such as *y* = (1.02)^{t}, *y* = (0.97)^{t}, *y* = (1.01)^{12t}, and *y* = (1.2)^{t/10}, and classify them as representing exponential growth and decay.

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

Earlier in this video series, students transformed quadratic expressions from standard form into factored form. There, the factored expressions are products of two sums, (*x* + *m*)(*x* + *n*), or two differences, (*x* – *m*)(*x* – *n*). Students continue that work in this video lesson, extending it to include expressions that can be rewritten as products of a sum and a difference, (*x* + *m*)(*x* – *n*).

Through repeated reasoning, students notice that when we apply the distributive property to multiply out a sum and a difference, the product has a negative constant term, but the linear term can be negative or positive (MP8). Students make use of the structure as they take this insight to transform quadratic expressions into factored form (MP7).

32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

Example: Identify percent rate of change in functions such as *y* = (1.02)^{t}, *y* = (0.97)^{t}, *y* = (1.01)^{12t}, and *y* = (1.2)^{t/10}, and classify them as representing exponential growth and decay.

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

In this video lesson, students encounter quadratic expressions without a linear term and consider how to write them in factored form.

Through repeated reasoning, students are able to generalize the equivalence of these two forms: (*x* + *m*)(*x* – *m*) and *x*^{2} – *m*^{2} (MP8). Then, they make use of the structure relating the two expressions to rewrite expressions (MP7) from one form to the other.

Students also consider why a difference of two squares (such as *x*^{2} – 25) can be written in factored form, but a sum of two squares (such as *x*^{2} + 25) cannot be, even though both are quadratic expressions with no linear term.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 32 : 32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

Example: Identify percent rate of change in functions such as *y* = (1.02)^{t}, *y* = (0.97)^{t}, *y* = (1.01)^{12t}, and *y* = (1.2)^{t/10}, and classify them as representing exponential growth and decay.

[MA2015] AL2 (9-12) 4 :

[MA2015] ALT (9-12) 4 :

[MA2019] AL1-19 (9-12) 6 :6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

*Example: Identify percent rate of change in functions such as *y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}*, and classify them as representing exponential growth or decay.* [MA2019] AL1-19 (9-12) 9 :

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2019] AL1-19 (9-12) 6 :

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

In this video lesson, students apply what they learned about transforming expressions into factored form to make sense of quadratic equations and persevere in solving them (MP1). They see that rearranging equations so that one side of the equal sign is 0, rewriting the expression in factored form, and then using the zero product property make it possible to solve equations that they previously could only solve by graphing. These steps also allow them to easily see—without graphing and without necessarily completing the solving process—the number of solutions that the equations have.

View Standards
**Standard(s): **
[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 13 :

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 6 : 6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

*Example: Identify percent rate of change in functions such as *y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}*, and classify them as representing exponential growth or decay.* [MA2019] AL1-19 (9-12) 9 :

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

This video lesson has two key aims. The first aim is to familiarize students with the structure of perfect-square expressions. Students analyze various examples of **perfect squares**. They apply the distributive property repeatedly to expand perfect-square expressions given in the factored form (MP8). The repeated reasoning allows them to generalize expressions of the form (*x* + *n*)^{2} as equivalent to *x*^{2} + 2*nx* + *n*^{2}.

The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7), which they will do in a future video lesson.

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 16 :

[MA2015] AL1 (9-12) 17 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 13 :

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] AL2 (9-12) 20 :

[MA2015] AL2 (9-12) 24 :

[MA2015] AL2 (9-12) 29 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 20 :

[MA2015] ALT (9-12) 24 :

[MA2015] ALT (9-12) 29 :

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 6 : 6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

*Example: Identify percent rate of change in functions such as *y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}*, and classify them as representing exponential growth or decay.* [MA2019] AL1-19 (9-12) 9 :

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [MA2019] AL1-19 (9-12) 11 :

16 ) Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1]

[MA2015] AL1 (9-12) 17 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2015] AL2 (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2015] AL2 (9-12) 24 :

24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]

[MA2015] AL2 (9-12) 29 :

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

Example: If the function *h*(*n*) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2015] ALT (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2015] ALT (9-12) 24 :

24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]

[MA2015] ALT (9-12) 29 :

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

Example: If the function *h*(*n*) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. **Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.**

[MA2019] AL1-19 (9-12) 15 : 15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. *Note: If *f* is a function and *x *is an element of its domain, then *f(x) *denotes the output of* f* corresponding to the input *x*.*

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. **Limit to linear, quadratic, exponential, and absolute value functions.**

In this video lesson, students return to some quadratic functions they have seen. They write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward (MP5).

The work in this lesson—writing equations, solving them, and interpreting the solutions in context—encourages students to reason quantitatively and abstractly (MP2).

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 16 :

[MA2015] AL1 (9-12) 17 :

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 21 :

[MA2015] AL2 (9-12) 27 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 13 :

Example: See *x*^{4} - *y*^{4} as (*x*^{2})^{2} - (*y*^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (*x*^{2} - *y*^{2})(*x*^{2} + *y*^{2}).

[MA2015] ALT (9-12) 21 :

[MA2015] ALT (9-12) 27 :

[MA2019] AL1-19 (9-12) 5 :

*Example: See *x^{4} - y^{4}* as *(x^{2})^{2} - (y^{2})^{2}*, thus recognizing it as a difference of squares that can be factored as *(x^{2} - y^{2})(x^{2} + y^{2}). [MA2019] AL1-19 (9-12) 9 :

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [MA2019] AL1-19 (9-12) 10 :

16 ) Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1]

[MA2015] AL1 (9-12) 17 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 21 :

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

[MA2015] AL2 (9-12) 27 :

27 ) Explain why the *x*-coordinates of the points where the graphs of the equations *y* = *f*(*x*) and *y* = *g*(*x*) intersect are the solutions of the equation *f*(*x*) = *g*(*x*); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f*(*x*) and/or *g*(*x*) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 13 :

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

[MA2015] ALT (9-12) 21 :

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

[MA2015] ALT (9-12) 27 :

27 ) Explain why the *x*-coordinates of the points where the graphs of the equations *y* = *f*(*x*) and *y* = *g*(*x*) intersect are the solutions of the equation *f*(*x*) = *g*(*x*); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where *f*(*x*) and/or *g*(*x*) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

9. Select an appropriate method to solve a quadratic equation in one variable.

10. Select an appropriate method to solve a system of two linear equations in two variables.

a. Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.

b. Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.

[MA2019] AL1-19 (9-12) 12 : 12. Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. **Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.**

[MA2019] AL1-19 (9-12) 14 : 14. Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane.

*Note: The graph of a relation often forms a curve (which could be a line).*

[MA2019] AL1-19 (9-12) 15 : 15. Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.

a. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. *Note: If *f* is a function and *x *is an element of its domain, then *f(x) *denotes the output of* f* corresponding to the input *x*.*

b. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. **Limit to linear, quadratic, exponential, and absolute value functions.**

[MA2019] AL1-19 (9-12) 19 : 19. Explain why the x-coordinates of the points where the graphs of the equations *y = f(x)* and *y = g(x)* intersect are the solutions of the equation *f(x) = g(x).*

a. Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate.

*Note: Include cases where *f(x)* is a linear, quadratic, exponential, or absolute value function and *g(x) *is constant or linear.*

[MA2019] AL1-19 (9-12) 20 : 20. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate.

This video lesson builds on the idea that both graphing and rewriting quadratic equations in the form of expression = 0 are useful strategies for solving equations. It also reinforces the ties between the zeros of a function and the horizontal intercepts of its graph, which students began exploring in an earlier unit.

Here, students learn that they can solve equations by rearranging them into the form expression = 0, graphing the equation *y* = expression, and finding the horizontal intercepts. They also notice that dividing each side of a quadratic equation by a variable is not reliable because it eliminates one of the solutions. As students explain why certain maneuvers for solving quadratic equations are acceptable and others are not, students practice constructing logical arguments (MP3).

View Standards
**Standard(s): **
[MA2015] AL1 (9-12) 17 :

[MA2015] AL1 (9-12) 32 :32 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a]

Example: Identify percent rate of change in functions such as *y* = (1.02)^{t}, *y* = (0.97)^{t}, *y* = (1.01)^{12t}, and *y* = (1.2)^{t/10}, and classify them as representing exponential growth and decay.

[MA2015] AL2 (9-12) 4 :

[MA2015] AL2 (9-12) 20 :

[MA2015] ALT (9-12) 4 :

[MA2015] ALT (9-12) 20 :20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2019] AL1-19 (9-12) 6 :6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.

b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.

*Example: Identify percent rate of change in functions such as *y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}*, and classify them as representing exponential growth or decay.* [MA2019] AL1-19 (9-12) 9 :

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form *(x - p)*^{2} *= q *that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as *x*^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [MA2019] AL1-19 (9-12) 11 :

17 ) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3]

[MA2015] AL1 (9-12) 32 :

b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b]

[MA2015] AL2 (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] AL2 (9-12) 20 :

20 ) Create equations and inequalities in one variable and use them to solve problems. *Include equations arising from linear and quadratic functions, and simple rational and exponential functions.* [A-CED1]

[MA2015] ALT (9-12) 4 :

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

[MA2015] ALT (9-12) 20 :

[MA2019] AL1-19 (9-12) 6 :

c. Use the properties of exponents to transform expressions for exponential functions.

9. Select an appropriate method to solve a quadratic equation in one variable.

11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. **Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.**

In this video lesson, students learn about the **zero product property**. They use it to reason about the solutions to quadratic equations that each have a quadratic expression in the factored form on one side and 0 on the other side. They see that when an expression is a product of two or more factors and that product is 0, one of the factors must be 0. Students make use of the structure of a quadratic expression in factored form and the zero product property to understand the connections between the numbers in the form and the *x*-intercepts of its graph (MP7).