5-3 Solving Quadratic Equations by Graphing and Factoring Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Find the x-intercept of each function. 1. f(x) = –3x + 9 3 Factor each expression. 3. 3x2 – 12x 3x(x – 4) 4. x2 – 9x + 18 (x – 6)(x – 3) 5. x2 – 49 (x – 7)(x + 7)

Objectives Solve quadratic equations by graphing or factoring. Determine a quadratic function from its roots.

A zero of a function is a value of the input x that makes the output f(x) equal zero. The zeros of a function are the x-intercepts. Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at right. These zeros are always symmetric about the axis of symmetry.

Example 1: Finding Zeros by using your graphing calculator Find the zeros of f(x) = x2 – 6x + 8 by using a graph and zero function on your calculator. The zeros are 2 and 4

Check It Out! Example 1 Find the zeros of g(x) = –x2 – 2x + 3 by using a graph and a table. Zeros are -3 and 1

You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Functions have zeros or x-intercepts. Equations have solutions or roots. Reading Math

Example 2A: Finding Zeros by Factoring Find the zeros of the function by factoring. f(x) = x2 – 4x – 12 x2 – 4x – 12 = 0 Set the function equal to 0. (x + 2)(x – 6) = 0 Factor: Find factors of –12 that add to –4. x + 2 = 0 or x – 6 = 0 Apply the Zero Product Property. x= –2 or x = 6 Solve each equation.

Find the zeros of the function by factoring. Example 2A Continued Find the zeros of the function by factoring. Check Substitute each value into original equation. x2 – 4x – 12 = 0 x2 – 4x – 12 = 0 (–2)2 – 4(–2) – 12 (6)2 – 4(6) – 12 36 – 24 – 12 4 + 8 – 12  

Example 2B: Finding Zeros by Factoring Find the zeros of the function by factoring. g(x) = 3x2 + 18x 3x2 + 18x = 0 Set the function to equal to 0. 3x(x+6) = 0 Factor: The GCF is 3x. 3x = 0 or x + 6 = 0 Apply the Zero Product Property. x = 0 or x = –6 Solve each equation.

Check Check algebraically and by graphing. Example 2B Continued Check Check algebraically and by graphing. 3x2 + 18x = 0 3x2 + 18x = 0 3(0)2 + 18(0) 3(–6)2 + 18(–6) 0 + 0 108 – 108   5 –10 –30 25

Find the zeros of the function by factoring. Check It Out! Example 2a Find the zeros of the function by factoring. f(x)= x2 – 5x – 6 x2 – 5x – 6 = 0 Set the function equal to 0. (x + 1)(x – 6) = 0 Factor: Find factors of –6 that add to –5. x + 1 = 0 or x – 6 = 0 Apply the Zero Product Property. x = –1 or x = 6 Solve each equation.

Check It Out! Example 2a Continued Find the zeros of the function by factoring. Check Substitute each value into original equation. x2 – 5x – 6 = 0 x2 – 5x – 6 = 0 (–1)2 – 5(–1) – 6 (6)2 – 5(6) – 6 36 – 30 – 6 1 + 5 – 6  

Find the zeros of the function by factoring. Check It Out! Example 2b Find the zeros of the function by factoring. g(x) = x2 – 8x x2 – 8x = 0 Set the function to equal to 0. x(x – 8) = 0 Factor: The GCF is x. x = 0 or x – 8 = 0 Apply the Zero Product Property. x = 0 or x = 8 Solve each equation.

Check It Out! Example 2b Continued Find the zeros of the function by factoring. Check Substitute each value into original equation. x2 – 8x = 0 x2 – 8x = 0 (0)2 – 8(0) (8)2 – 8(8) 64 – 64 0 – 0  

Quadratic expressions can have one, two or three terms, such as –16t2, –16t2 + 25t, or –16t2 + 25t + 2. Quadratic expressions with two terms are binomials. Quadratic expressions with three terms are trinomials. Some quadratic expressions with perfect squares have special factoring rules.

Example 4A: Find Roots by Using Special Factors Find the roots of the equation by factoring. 4x2 = 25

Example 4 Continued Check Graph the related function f(x) = 4x2 – 25 on a graphing calculator. The function appears to have zeros at and . –3 30 3 10

Find the roots of the equation by factoring. Check It Out! Example 4b Find the roots of the equation by factoring. 25x2 = 9 25x2 – 9 = 0 Rewrite in standard form. (5x + 3)(5x – 3) = 0 Factor the difference of squares. 5x + 3 = 0 or 5x – 3 = 0 Apply the Zero Product Property. x = or x = Solve each equation.

Check It Out! Example 4b Continued Check Graph the related function f(x) = 25x2 – 9 on a graphing calculator. The function appears to have zeros at and . –1 10 1

If you know the zeros of a function, you can work backward to write a rule for the function

Example 5: Using Zeros to Write Function Rules Write a quadratic function in standard form with zeros 4 and –7. x = 4 or x = –7 Write the zeros as solutions for two equations. x – 4 = 0 or x + 7 = 0 Rewrite each equation so that it equals 0. (x – 4)(x + 7) = 0 Apply the converse of the Zero Product Property to write a product that equals 0. x2 + 3x – 28 = 0 Multiply the binomials. f(x) = x2 + 3x – 28 Replace 0 with f(x).

Check Graph the function Example 5 Continued 10 –35 –10 Check Graph the function f(x) = x2 + 3x – 28 on a calculator. The graph shows the original zeros of 4 and –7.

Write a quadratic function in standard form with zeros 5 and –5. Check It Out! Example 5 Write a quadratic function in standard form with zeros 5 and –5. x = 5 or x = –5 Write the zeros as solutions for two equations. x + 5 = 0 or x – 5 = 0 Rewrite each equation so that it equals 0. (x + 5)(x – 5) = 0 Apply the converse of the Zero Product Property to write a product that equals 0. x2 – 25 = 0 Multiply the binomials. f(x) = x2 – 25 Replace 0 with f(x).

Check It Out! Example 5 Continued 10 8 –30 –8 Check Graph the function f(x) = x2 – 25 on a calculator. The graph shows the original zeros of 5 and –5.

Note that there are many quadratic functions with the same zeros Note that there are many quadratic functions with the same zeros. For example, the functions f(x) = x2 – x – 2, g(x) = –x2 + x + 2, and h(x) = 2x2 – 2x – 4 all have zeros at 2 and –1. –5 –7.6 5 7.6

Lesson Quiz: Part I Find the zeros of each function. 1. f(x)= x2 – 7x 0, 7 2. f(x) = x2 – 9x + 20 4, 5 Find the roots of each equation using factoring. 3. x2 – 10x + 25 = 0 5 4. 7x = 15 – 2x2 –5,

Lesson Quiz: Part II 5. Write a quadratic function in standard form with zeros 6 and –1. Possible answer: f(x) = x2 – 5x – 6 6. A rocket is launched from ground level with an initial vertical velocity of 176 ft/s. After how many seconds with the rocket hit the ground? after 11 s