The Quadratic Formula.

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242

Lesson 9.8

Warm UpEvaluate for x = –2, y = 3, and z =

–1. 6 1. x2 2. xyz

3. x2 – yz 4. y – xz 4

5. –x 6. z2 – xy

7 1

7 2

California Standards

19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

In the previous lesson, you completed the square to solve quadratic

equations. If you complete the square of ax2 + bx + c = 0, you can derive

the Quadratic Formula.

What Does The Formula Do ?

The Quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise.The formula states that for a quadratic equation of the form :ax2 + bx + c = 0 The roots of the quadratic equation are given by :

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242

Example 1

Use the quadratic formula to solve the equation :x 2 + 5x + 6= 0Solution:x 2 + 5x + 6= 0a = 1 b = 5 c = 6

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242

12)614(55 2

x

2)24(255

x

215

x

215

215

xorx

x = - 2 or x = - 3

These are the roots of the equation.

Example 2

Use the quadratic formula to solve the equation :8x 2 + 2x - 3= 0

Solution:8x 2 + 2x - 3= 0a = 8 b = 2 c = -3

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242

82)384(22 2

x

16)96(42

x

161002

x

16102

16102

xorx

x = ½ or x = - ¾ These are the roots of the equation.

Example 3Use the quadratic formula to solve the equation :8x 2 - 22x + 15= 0Solution:8x 2 - 22x + 15= 0a = 8 b = -22 c = 15

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242

82)1584()22()22( 2

x

16)480(484(22

x

16422

x

16222

16222

xorx

x = 3/2 or x = 5/4 These are the roots of the equation.

Because the Quadratic Formula contains a square root, the solutions may be irrational. You can give the exact solution by leaving the square root in your answer, or you can approximate the solutions.

1. Solve x2 + x = 12 by using the Quadratic Formula.

2. Solve –3x2 + 5x = 1 by using the Quadratic Formula.

3. Solve 8x2 – 13x – 6 = 0. Use at least 2 different methods.

Lesson Quiz

3, –4

= 0.23, ≈ 1.43