5 – 2: Solving Quadratic 5 – 2: Solving Quadratic Equations by FactoringEquations by Factoring

Objective: CA 8: Students solve and graph quadratic equations by factoring,

completing the square, or using the quadratic formula.

You can use factoring to write a trinomial as a product of two binomials.

To factor 2x bx c find integers m and n such that

2

2

x bx c x m x n

x m n x mn

So, the sum of m and n must equal b and the product of m and n must equal c

Example 1

Factoring a Trinomial of the form 2x bx c

Factor 2 12 28x x

2 12 28

where 28 and 12.

x x x m x n

mn m n

Factor -28 Sum of factors -12

-1 x 28 – 1 + 28 = 27

-2 x 14 - 2 + 14 = 12

-4 x 7 - 4 + 7 = 3

1 x – 28 1 - 28 = - 27

2 x – 14 2 – 14 = - 12

4 x –7 4 – 7 = - 3

The table shows that m = 2 and n = -14

2 12 28 2 14x x x x

2

2

2

To factor , when 1,

find k, l, m, and n such that:

ax bx c a

ax bx c kx m lx n

klx kn lm x mn

Therefore k, and l must be factors of a, and m, and n must be factors of c.

Example 2

Factoring a Trinomial of the form

Factor

2ax bx c 23 17 10x x

23 17 10x x kx m lx n

Where k and l are factors of 3 and m and n are (negative) factors of 10

The correct factorization is

23 17 10 3 2 5x x x x

Special Factoring Patterns

Difference of Squares 2 2a b a b a b

2 9 3 3x x x

Perfect Trinomial Squares

22 22a ab b a b

22 12 36 6x x x

22 22a ab b a b

22 8 16 4x x x

Example 3

Factoring with Special Patterns

Factor the quadratic expressions

22 24 25 2 5x x Difference of Squares

2 5 2 5x x

29 24 16y y

23 4y Prefect Square Trinomial

249 14 1r r

27 1r Prefect Square Trinomial

2 27 2 7 1 1r r

2 23 2 3 4 4y y

A monomial is an expression with only one term. As a first step in factoring, you should check to see whether the terms have a common monomial factor.

Example 4

Factor the quadratic expression.25 20x

2 25 20 5 4

5 2 2

x x

x x

Factor the quadratic expression.26 15 9p p

2 26 15 9 3 2 5 3p p p p

3 2 3 1p p

You can use factoring to solve certain quadratic equations. A quadratic equation in one variable can be written in the form 2 0ax bx c where a 0. This is called the Standard form of the equation. If the left side of

2 0ax bx c

can be factored, then the equation can be solved using the zero product property.

Zero Product Property

Let A and B be real numbers of algebraic expressions. If AB = 0, then A = 0 or B = 0.

Example 5

Solving Quadratic Equations

Solve 2 3 18 0x x

2 3 18 6 3x x x x

6 0 or 3 0

6 3

x x

x x

The solutions are –6 and 3. Check the solutions in the original equation.

Solve: 22 17 45 3 5t t t

2

2

2

2

2 20 50 0

2 10 25 0

10 25 0

5 0

5

t t

t t

t t

t

t

The solution is 5. Check the solution in the original equation.

Home work page 260, 23 – 31 odd, 35 – 43 odd, 47 – 51 odd, 57 – 69 odd.