COMPLETING THE SQUARE

Not every quadratic expression is a perfect square trinomial. Completing the square is the process of finding the constant to add to

x² + bx

to create a perfect square trinomial.

Understand the Process of Completing the Square

The model below depicts the process of completing the square.

To create a perfect square trinomial, add (b/2)² to the variable expression.

x² + bx + (b/2)² = (x + b/2)²

Example 1 :

Write x² + 6x + 7 = 0 in the form (x + p)² = q.

Solution :

Write the original equation.

x² + 6x + 7 = 0

Isolate the variable expression.

x² + 6x = -7 -----(1)

Determine the constant needed to complete the square.

Comparing x² + bx and x² + 6x, we get

b = 6

So,

(b/2)² = (6/2)² = 3² = 9

In (1), we have to add 9 to each side.

x² + 6x + 9 = -7 + 9

Write the left side of the equation as a perfect square.

(x + 3)² = 2

Hence, the equation x² + 6x + 7 = 0 can be written as 

(x + 3)² = 2

Use Square Roots to Solve Quadratic Equations

Example 2 :

Solve the following quadratic equation using square root :

x² + 12x + 36 = 49

Solution :

Write the original equation.

x² + 12x + 36 = 49

Recognize that the quadratic equation is a perfect square trinomial.

x² + 2(6)(x) + 6² = 49

Factor the perfect square trinomial.

(x + 6)² = 49

Take the square root on each side of the equation.

√(x + 6)² = ± √49

x + 6 = ± 7

x + 6 = -7  or  x + 6 = 7

x = -13  or  x = 1

Solving a Quadratic Equation by Completing the Square

Example 3 :

Solve the following quadratic equation by completing the square :

x² - 8x - 9 = 0

Solution :

Write the original equation.

x² - 8x - 9 = 0

Isolate the variable expression.

x² - 8x = 9 -----(1)

Determine the constant needed to complete the square.

Comparing x² + bx and x² - 8x, we get

b = -8

So,

(b/2)² = (-8/2)² = (-4)² = 16

In (1), we have to add 16 to each side.

x² - 8x + 16 = 9 + 16

x² - 8x + 16 = 25

Write the left side of the equation as a perfect square.

(x - 4)² = 25

Take the square root on each side of the equation.

√(x - 4)² = ± √25

x - 4 = ±5

x - 4 = -5  or  x - 4 = 5

x = -1  or  x = 9

Write a Quadratic Equation in Vertex Form

Example 4 :

Write the following quadratic equation in vertex form and graph it :

y = -x² - 2x + 3

What is the maximum or minimum value of the graph of the equation ?

Solution :

Write the original equation.

y = -x² - 2x + 3

Factor out the x² coefficient, -1.

y = -1(x² + 2x + 3)

= -1(x² - 2 ⋅ x ⋅ 1 - 3)

= -1(x² - 2 ⋅ x ⋅ 1 - 3)

= -1(x² - 2 ⋅ x ⋅ 1 + 1² - 1² - 3)

= -[(x - 1)² - 1² - 3]

= -[(x - 1)² - 1 - 3]

= -[(x - 1)² - 4]

= -(x - 1)² + 4

Hence, the vertex form of the equation y = -x² - 2x + 3 is

y = -(x - 1)² + 4

Vertex :

The vertex of the parabola is (1, 4).

Graph :

In the given equation y = -x² - 2x + 3, the sign of x² is negative.

So, its graph is a parabola that opens downward.

The graph of the given quadratic equation has a maximum of y = 4 at x = 1.

Complete the Square to Solve a Real-World Problem

Example 5 :

Alex plans to create rectangular shaped garden. He has 340 m of fencing available for the garden's perimeter and wants it to have an area of 6000 m². What dimensions should Alex use ?

Solution :

Let x and y be the length and width of the garden respectively.

Given : Perimeter = 340.

So, we have

2x + 2y = 340

Divide each side by 2.

x + y = 170

Solve for y.

y = 170 - x

Alex wants the area to be 6000 m².

Write this as an equation.

A = xy

6000 = x(170 - x)

6000 = 170x - x²

x² - 170x = -6000 -----(1)

Determine the constant needed to complete the square.

Comparing x² + bx and x² - 170x, we get

b = -170

So,

(b/2)² = (-170/2)² = (-85)² = 7225

In (1), we have to add 7225 to each side.

x² - 170x + 7225 = -6000 + 7225

Write the left side of the equation as a perfect square.

(x - 85)² = 1225

Take the square root on each side of the equation.

√(x - 85)² = ±√1225

x - 85 = ±35

x - 85 = -35  or  x - 85 = 35

x = 50  or  x = 110

When x = 50,

y = 170 - 50

y = 120

When x = 110,

y = 170 - 110

y = 60

In each case, there is 340 m of fencing used.

Likewise, the area is 6000 m².

Hence, Alex should make two sides of the garden 120 m long and the other two sides 50 m long.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.