Lesson 2-3 The Quadratic Equation

Objective: To learn the various ways to solve quadratic

equations, including factoring, completing the square and the

quadratic formula.

Quadratic Equations A quadratic equation in x is an equation that can

be written in the general form ax2 + bx + c = 0 where a, b, and c are real numbers We can solve by several methods: • By Factoring and setting each factor

equal to 0 • Extracting Square Roots • Completing the Square • Using the Quadratic Formula

Solving by Factoring Factor ax2 + bx + c =(Ax+B)(Cx+D) = 0

• Set each factor = 0. (Ax+B) = 0, (Cx+D) = 0

• Solve for x

Solving by Factoring x2 -7x + 12=0 find factors of 12 that

add to -7

(x – 3)(x – 4) = 0

x-3 = 0 x–4 =0 x = 3 x = 4

Solving by Factoring 2x2 – 7x -15 = 0 (2x + 3)( x- 5) = 0

2x + 3 = 0 x – 5 = 0 2x = -3 x = x = 5

23

Solving by Factoring 4x2 – 3x = 0

x(4x – 3) = 0

x = 0 4x -3 = 0 4x = 3 x =

43

Practice 4x2 – x = 0

3x2 – 11x -4 = 0

Solving by Completing the SquareSolve the following

equation by completing the square:

Step 1: Move quadratic term, and linear term to left side of the equation

2 8 20 0x x

2 8 20x x

Solving by Completing the Square

Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

2 8 =20 + x x 21 ( ) 4 then square it, 4 16

28

2 8 2016 16x x

2

2

b

Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

2 8 2016 16x x

Solving by Completing the Square

36)4)(4( xx

36)4( 2 x

Solving by Completing the Square

Step 4: Take the square root of each side

2( 4) 36x

( 4) 6x

Solving by Completing the Square

Step 5: Set up the two possibilities and solve

4 64 6 an

d 4 6

10 and 2 x=

xx xx

Completing the Square-Example #2Solve the following

equation by completing the square:

Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.

22 7 12 0x x

22 7 12x x

Solving by Completing the Square

Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.

2

2

2

2 7

22 2 2

7 12

72

=-12 +

6

x x

x x

xx

21 7 7 49( ) then square it, 2 62 4 4 1

7

2 49 4916 1

7 62 6

x x

Solving Completing the Square

Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

2

2

2

7 627 96 494 16 16

7 474

49 4916 1

16

6x x

x

x

Solving by Completing the Square

Step 4: Take the square root of each side

27 47( )4 16

x

7 47( )4 47 474 47 47

4

x

ix

ix

Solving by Completing the Square

2

2

2

2

2

1. 2 63 0

2. 8 84 0

3. 5 24 0

4. 7 13 0

5. 3 5 6 0

x x

x x

x x

x x

x x

Try the following examples. Do your work on your paper and then check your answers.

1. 9,72.(6, 14)3. 3,8

7 34.2

5 475.6

i

i

Warm up Solve by factoring:

x2 + 5x +6=0

2x2 + 9x – 18 = 0

3x2 +x =0

Taking Square Roots x2 – 3 = 0 (can’t factor) x2 = 3 take the square root of both

sides x = √3 or x = - √3

x = ±√3

Taking Square Roots x2 + 9 = 0 x2 = -9

x = √-9 x = ±3i

Taking Square Roots 2(x – 1)2 – 4 = 0 +4 +4 2(x – 1)2 = 4 2 2

(x – 1)2 = 2 take the sq. root of each side x – 1 = ±√2 +1 +1 x = 1±√2

Practice 5x2 + 13 = 0

(2x – 7)2 – 5 = 0

The Quadratic Formula2 42

b b acxa

Songx equals negative bplus or minus the square rootof b squared minus 4acall over 2a

Quadratic FormulaLet’s look at an example.3x2 - 4x + 3 = 0a = ?b = ?c = ?

a = 3b = -4c = 3

Quadratic FormulaNow let’s plug it in.b = -4, so -b = -(-4) = 4

4 ( 4)2 4(3)(3)2(3)

Quadratic Formula

Simplify

4 16 366

4 16 366

Quadratic Formula

Find the zeros ofr2 - 7r -18 = 0

27 ( 7) 4(1)( 18)2(1)

r

Quadratic FormulaSimplify

7 49 ( 72)2

7 121 7 112 2

r

Quadratic Formula

Now let’s examine our solution.

We can break this into two equations.

7 112

r

Quadratic Formula Now we can get our two

solutions.7 11 18 9

2 27 11 4 2

2 2

r

r

Quadratic Formula x2 – 8x = -10

4x2 -2x +1 = 0