Exercises. Example 1. Example 2. Chapter 9. 20. Glencoe Algebra 1. Study Guide and Intervention. Solving Quadratic Equations by Completing the Square ...
NAME ______________________________________________ DATE______________ PERIOD _____
9-3
Study Guide and Intervention Solving Quadratic Equations by Completing the Square
Find the Square Root the square root of each side.
An equation such as x2 ⫺ 4x ⫹ 4 ⫽ 5 can be solved by taking
Example 1
Example 2
Solve x2 ⫺ 2x ⫹ 1 ⫽ 9. Round to the nearest tenth if necessary. x2 ⫺ 2x ⫹ 1 ⫽ 9 (x ⫺ 1)2 ⫽ 9 兹苶 (x ⫺ 1苶 )2 ⫽ 兹苶 9 ! x ⫺ 1 ! ⫽ 兹苶9 x ⫺ 1 ⫽ ⫾3 x ⫺ 1 ⫹ 1 ⫽ ⫾3 ⫹ 1 x⫽1⫾3
Solve x2 ⫺ 4x ⫹ 4 ⫽ 5. Round to the nearest tenth if necessary. x2 ⫺ 4x ⫹ 4 ⫽ 5 (x ⫺ 2)2 ⫽ 5 兹苶 (x ⫺ 2苶 )2 ⫽ 兹苶 5 ! x ⫺ 2 ! ⫽ 兹苶5 x ⫺ 2 ⫽ ⫾兹苶 5 x ⫺ 2 ⫹ 2 ⫽ ⫾兹苶 5⫹2 5 x ⫽ 2 ⫾ 兹苶
x ⫽ 1 ⫹ 3 or ⫽4
Use a calculator to evaluate each value of x.
x⫽1⫺3 ⫽ ⫺2
x ⫽ 2 ⫹ 兹5 苶 or ⬇ 4.2
The solution set is {⫺2, 4}.
x ⫽ 2 ⫺ 兹5 苶 ⬇ ⫺0.2
The solution set is {⫺0.2, 4.2}. Exercises Solve each equation by taking the square root of each side. Round to the nearest tenth if necessary. 2. m2 ⫹ 12m ⫹ 36 ⫽ 1
NAME ______________________________________________ DATE______________ PERIOD _____
9-3
Study Guide and Intervention
(continued)
Solving Quadratic Equations by Completing the Square Complete the Square
Since few quadratic expressions are perfect square trinomials, the method of completing the square can be used to solve some quadratic equations. Use the following steps to complete the square for a quadratic expression of the form ax2 ⫹ bx. b 2
Step 1
Find ᎏ .
Step 2
Find ᎏ .
Step 3
Add ᎏ
冢 b2 冣2
冢 b2 冣2 to ax 2 ⫹ bx.
Example
Solve x2 ⫹ 6x ⫹ 3 ⫽ 10 by completing the square.
x2 ⫹ 6x ⫹ 3 ⫽ 10 x2
⫹ 6x ⫹ 3 ⫺ 3 ⫽ 10 ⫺ 3 x2 ⫹ 6x ⫽ 7 x2
⫹ 6x ⫹ 9 ⫽ 7 ⫹ 9 (x ⫹
3)2
⫽ 16
x ⫹ 3 ⫽ ⫾4 x ⫽ ⫺3 ⫾ 4 x ⫽ ⫺3 ⫹ 4 or ⫽1
Original equation Subtract 3 from each side. Simplify.
冢6冣
Since ᎏ 2 Factor
x2
2
⫽ 9, add 9 to each side.
⫹ 6x ⫹ 9.
Take the square root of each side. Simplify.
x ⫽ ⫺3 ⫺ 4 ⫽ ⫺7
Exercises Solve each equation by completing the square. Round to the nearest tenth if necessary. 1. t2 ⫺ 4t ⫹ 3 ⫽ 0