COMPLETING THE SQUARE

About "Completing the Square"

Completing the Square :

Not every quadratic expression is a perfect squaree 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 : 

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)=  (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 : 

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

Hence, the solutions are -13 and 1. 

Solving a Quadratic Equation by Completing the Square

Example : 

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

Hence, the solutions are -1 and 9.

Write a Quadratic Equation in Vertex Form

Example : 

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)

y  =  -1(x2 - 2 ⋅ x ⋅ 1 - 3)

y  =  -1(x2 - 2 ⋅ x ⋅ 1 - 3)

y  =  -1(x2 - 2 ⋅ x ⋅ 1 + 12 - 12 - 3)

y  =  - [(x - 1)- 12 - 3]

y  =  - [(x - 1)- 1 - 3]

y  =  - [(x - 1)- 4]

y  =  - (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 :

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.  

After having gone through the stuff given above, we hope that the students would have understood, "Completing the Square". 

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