Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

A trinomial (an expression with 3 terms) in standard form (ax2 +bx + c) can be factored by finding factors that multiply to equal c and add to equal b.

*Always check for a GCF first.

f(x) = x2 – 4x – 12

(x + 2)(x – 6)

c = -12, factors that multiply to equal -12

b = -4, add the factors and see which one

equals -4

1 and -12 1 + (-12) = -11

-1 and 12 -1 + 12 = 11

2 and -6 2 + (-6) = -4

-2 and 6 -2 + 6 = 4

3 and -4 3 + (-4) = -1

-3 and 4 -3 + 4 = 1

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

f(x)= x2 – 5x – 6

Find the zeros of the function by factoring.

x2 – 5x – 6 = 0

(x + 1)(x – 6) = 0

x + 1 = 0 x – 6 = 0

x = –1 or x = 6

Set the function equal to 0.

Factor: Find factors of –6 that add to –5.

Set each parenthesis equal to zero

Solve each equation.

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

Find the roots of the equation by factoring.

Example 4B: Find Roots by Using Special Factors

4x2 = 12x + 16

Rewrite in standard form.

Factor. The GCF is 4.

Factor the trinomial

Set each parenthesis equal to zero

Solve each equation.

4x2 – 12x – 16 = 0

4(x2 – 3x – 4) = 0

x – 4 = 0 x + 1 = 0

x = 4 x = -1

4(x – 4)(x + 1) = 0

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

x2 – 4x = –4

Find the roots of the equation by factoring.

x2 – 4x + 4 = 0

(x – 2)(x – 2) = 0

x – 2 = 0 x – 2 = 0

x = 2 x = 2

Rewrite in standard form.

Set each parenthesis equal to zero.

Solve each equation.

Factor the perfect-square trinomial.

Check It Out! Example 4a

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

Write a quadratic function in standard form with zeros 4 and –7.

Write the zeros as solutions for two equations.

Rewrite each equation so that it equals 0.

These two equations will represent the parenthesis had you factored the function.

Multiply the binomials.

x = 4 or x = –7

x – 4 = 0 or x + 7 = 0

(x – 4)(x + 7)

x2 + 3x – 28

f(x) = x2 + 3x – 28 Name the function.

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

Example 5 Continued

Check Graph the function f(x) = x2 + 3x – 28 on

a calculator. The graph shows the original zerosof 4 and –7.

10

10

–35

–10

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

Determine the equation of a quadratic function with a double root at x = -1

x = -1 x = –1

x + 1 = 0 x + 1 = 0

(x + 1)(x + 1)

x2 +2x + 1

g(x) = x2 +2x + 1

Write the zeros as solutions for two equations.

Rewrite each equation so that it equals 0.

Multiply the binomials.

Name the function.

These two equations will represent the parenthesis had you factored the function.