**Completing the Square Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on completing the square.

Before look at the worksheet, if you would like to know the stuff related to completing the square,

**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 ?

**Problem 1 :**

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

**Solution : **

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 = 6

So, (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

(x + 3)^{2} = 2

**Problem 2 :**

Solve the following quadratic equation using square root :

x^{2} + 12x + 36 = 49

**Solution : **

Write the original equation.

x^{2} + 12x + 36 = 49

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

Hence, the solutions are -13 and 1.

**Problem 3 :**

Solve the following quadratic equation by completing the square :

x^{2} - 8x - 9 = 0

**Solution : **

Write the original equation.

x^{2} - 8x - 9 = 0

Isolate the variable expression.

x^{2} - 8x = 9 -----(1)

Determine the constant needed to complete the square.

Comparing x^{2} + bx and x^{2} - 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.

x^{2} - 8x + 16 = 9 + 16

x^{2} - 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.

**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 ?

**Solution : **

Write the original equation.

y = - x^{2} - 2x + 3

Factor out the x^{2} coefficient, -1.

y = -1(x^{2} + 2x + 3)

y = -1(x^{2} - 2 ⋅ x ⋅ 1 - 3)

y = -1(x^{2} - 2 ⋅ x ⋅ 1 - 3)

y = -1(x^{2} - 2 ⋅ x ⋅ 1 + 1^{2} - 1^{2} - 3)

y = - [(x - 1)^{2 }- 1^{2} - 3]

y = - [(x - 1)^{2 }- 1 - 3]

y = - [(x - 1)^{2 }- 4]

y = - (x - 1)^{2} + 4

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

y = - (x - 1)^{2} + 4

**Vertex :**

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

**Graph :**

In the given equation y = -x^{2} - 2x + 3, the sign of x^{2 }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.

**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 ?

**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 m^{2}.

Write this as an equation.

A = xy

6000 = x(170 - x)

6000 = 170x - x^{2}

x^{2} - 170x = -6000 -----(1)

Determine the constant needed to complete the square.

Comparing x^{2} + bx and x^{2} - 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.

x^{2} - 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 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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