4.7 Complete the Square

Completing the square

The process that allows you to write an expression of the form x² + bx as the square of a binomial

COMPLETING THE SQUARE

Words To complete the square for the expression x² + bx, add .

Algebra x² + bx + _{____}=

= _{_____}

Example 1

Find the value of c that makes x² + 12x + c a perfect square trinomial. Then write the expression as the square of a binomial.

Solution

1. Find half the coefficient of x. = _6_

2. Square the result of Step 1. _6²_ = _36_

3. Replace c with the result of Step 2. __x² + 12x +36_

The trinomial x² + 12x + c is a perfect square when c = _36_. Then

_x² + 12x + 36_ = (_x + 6_)(_x + 6_) = (_x +6_)².

Example 2

Solve ax² + bx + c = 0 when a = 1

Solve x2 - 10x + 13 = 0 by completing the square.

Solution

x^2 – 10x +13 = 0	Write original equation.
_x^2 – 10x_ = _–13_	Write left side in the form x^2 + bx.
_x^2 – 10x + 25_ = _–13 +25_	Complete the square.
_(x - 5)^2_ = _12_	Write left side as a binomial squared.
_x –5_ = _____	Take square roots of each side.
x = _______	Solve for x.
x = _______	Simplify.

The solutions are _{____} and _______ .

Example 3

Solve ax² + bx + c = 0 when a ¹ 1

Solve 3x² - 12x + 27 = 0 by completing the square.

Solution

3x^2 - 12x + 27 = 0	Write original equation.
_x^2 - 4x +9_ = _0_	Divide each side by the coefficient of x^2.
_x^2 - 4x_ = _-9_	Write left side in the form x^2 + bx.
_x^2 - 4x + 4_ = _-9 + 4_	Complete the square.
_(x -2)^2_ = _-5_	Write left side as binomial squared.
_x -2 = ______	Take square roots of each side.
x = __2 + i__	Write in terms of the imaginary unit i.

The solutions are __2 + i__ and _2 - i___ .

Example 4

Write a quadratic function in vertex form

Write y = x² + 14x + 44 in vertex form. Then identify the vertex.

Solution

y = x^2 + 14x + 44	Write original function.
y + _49_ = (x^2 + 14x + _49_) + 44	Complete the square.
y + _49_ = (_x + 7_)^2 + 44	Write as a binomial squared.
y = _(x + 7)^2 - 5_	Solve for y.

The vertex form of the function is y = __(x + 7)² - 5_. The vertex is (_-7_, _-5_).