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The Quadratic Formula.
a
acbbx
2
42
Lesson 9.8
Warm Up
Evaluate 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 :
a
acbbx
2
42
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
a
acbbx
2
42
12
)614(55 2
x
2
)24(255 x
2
15x
2
15
2
15
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
a
acbbx
2
42
82
)384(22 2
x
16
)96(42 x
16
1002x
16
102
16
102
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= 0
Solution:
8x 2 - 22x + 15= 0a = 8 b = -22 c = 15
a
acbbx
2
42
82
)1584()22()22( 2
x
16
)480(484(22 x
16
422x
16
222
16
222
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
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