  ## Solve By Completing the Square

Solve each equation by Completing the Square.

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Completing the Square Lesson Brush up on your knowledge of the techniques needed to solve problems on this page.

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## COMPLETING THE SQUARE WORKSHEET

Problem 1 :

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

Problem 2 :

Solve the following quadratic equation using square root :

x 2  + 12x + 36 = 49

Problem 3 :

Solve the following quadratic equation by completing the square :

x 2  - 8x - 9 = 0

Problem 4 :

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

y = -x 2  - 2x + 3

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

Problem 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 m 2 . What dimensions should Alex use ? Write the original equation.

x 2  + 6x + 7 = 0

Isolate the variable expression.

x 2  + 6x = -7 -----(1)

Determine the constant needed to complete the square.

Comparing  x 2  + bx and  x 2  + 6x, we get

(b/2) 2  = (6/2) 2  = 3 2  = 9

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

x 2  + 6x + 9 = -7 + 9

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

(x + 3) 2  = 2

Hence, the equation  x 2  + 6x + 7  =  0 can be written as

Recognize that the quadratic equation is a perfect square trinomial.

x 2  + 2(6)(x)  +  6 2  = 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

x 2  - 8x = 9 -----(1)

Comparing  x 2  + bx and  x 2  - 8x, we get

(b/2) 2  = (-8/2) 2  = (-4) 2  = 16

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

x 2  - 8x + 16 = 9 + 16

x 2  - 8x + 16 = 25

(x - 4) 2  = 25

√(x - 4) 2  = ± √25

x - 4 = -5  or  x - 4 = 5

x = -1  or  x = 9

Factor out the x 2  coefficient, -1.

y = -1(x 2  + 2x + 3)

= -1( x 2  - 2 ⋅ x ⋅ 1 - 3 )

= -1( x 2  - 2 ⋅ x ⋅ 1 + 1 2  - 1 2  -  3 )

= -[(x - 1) 2  - 1 2  - 3]

= -[(x - 1) 2  - 1 - 3]

= -[(x - 1) 2  - 4]

= -(x - 1) 2  + 4

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

y = -(x - 1) 2  + 4

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

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

So, its graph is a parabola that opens down. The graph of the given quadratic equation has a maximum of  y = 4 at x = 1.

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 m 2 .

Write this as an equation.

6000 = x(170 - x)

6000 = 170x - x 2

x 2  - 170x = -6000 -----(1)

Comparing  x 2  + bx and  x 2  - 170x, we get

(b/2) 2  = (-170/2) 2  = (-85) 2  = 7225

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

x 2  - 170x + 7225 = -6000 + 7225

(x - 85) 2  = 1225

√(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

When x = 110,

y = 170 - 110

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

Likewise, the area is 6000 m 2 .

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

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Subject: Mathematics

Age range: 14 - 18

Resource type: Worksheet/Activity Last updated

15 January 2012 Creative Commons "Sharealike"

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