Completing the square?

Question

Express in completed square form: a) x^2+3x-7 b)10-4x-x^2

Answer 1

a)

Given function f(x) = x²+3x-7

x² + 3x - 7

General Completed square form is a(x+d)² + e

To change the expression into a Completed square form add and suntract (half the x coefficient)²

Here x coefficient = 3. so, (half the x coefficient)² = (3/2)²= 9/4.

Add 9/4 to each side.

x² + 3x - 7 + 9/4 - 9/4

x² + 3x + 9/4 - 9/4 - 7

x² + 3x + (3/2)² - (28 + 9)/4

(x + 3/2)² - 37/4

The Completed square form is (x + 3/2)² - 37/4

Answer 2

b)

Given function f(x) = 10 - 4x - x²

10 - 4x - x²

10 - ( 4x + x² )

General Completed square form is a(x+d)² + e

To change the expression into a Completed square form add and subtract (half the x coefficient)²

Here x coefficient = 4. so, (half the x coefficient)² = (4/2)² = 2² = 4.

Add 2 to each side.

10 - ( 4x + x² ) + 4 - 4

10 + 4 - ( 4x + x² ) - 4

10 + 4 - ( 4x + x² + 4 )

10 + 4 - ( x² + 4x + 2² )

14 - (x + 2)²

The Completed square form is 14 - (x + 2)²