practice worksheet completing the square

Solve By Completing the Square

Solve each equation by Completing the Square.

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• Equation: x^2 + 5x + 6 = 0 Show answer | Show step-by-step Answer: x = -3, -2 Hide answer | Show step-by-step
• Equation: x^2 + 2x + 1 = 0 Show answer | Show step-by-step Answer: x = -1 Hide answer | Show step-by-step
• Equation: x^2 + 3x - 10 = 0 Show answer | Show step-by-step Answer: x = -5, 2 Hide answer | Show step-by-step
• Equation: m^2 - 4m - 21 = 0 Show answer | Show step-by-step Answer: m = -3, 7 Hide answer | Show step-by-step
• Equation: 3k^2 + 18k + 15 = 0 Show answer | Show step-by-step Answer: k = -5, -1 Hide answer | Show step-by-step
• Equation: 2n^2 - 22n + 60 = 0 Show answer | Show step-by-step Answer: n = 5, 6 Hide answer | Show step-by-step
• Equation: 4y^2 + 8y - 21 = 0 Show answer | Show step-by-step Answer: y = -3.5, 1.5 Hide answer | Show step-by-step
• Equation: 8y^2 + 10y - 3 = 0 Show answer | Show step-by-step Answer: y = -1.5, 0.25 Hide answer | Show step-by-step
• Equation: 50k^2 - 20k - 6 = 0 Show answer | Show step-by-step Answer: k = -0.2, 0.6 Hide answer | Show step-by-step
• Equation: 100x^2 - 100x + 9 = 0 Show answer | Show step-by-step Answer: x = 0.1, 0.9 Hide answer | Show step-by-step
• Equation: 10t^2 = t + 2 Show answer | Show step-by-step Answer: t = -0.4, 0.5 Hide answer | Show step-by-step
• Equation: 18m^2 + 6m + 6 = 6m^2 + 30m - 3 Show answer | Show step-by-step Answer: m = 0.5, 1.5 Hide answer | Show step-by-step
• Equation: p^2 - 0.75p + 0.125 = 0 Show answer | Show step-by-step Answer: p = 0.25, 0.5 Hide answer | Show step-by-step
• Equation: 2x^2 = x^2 + 1.75 - 3x Show answer | Show step-by-step Answer: x = -3.5, 0.5 Hide answer | Show step-by-step
• Equation: x^2 = 0.6x - 0.08 Show answer | Show step-by-step Answer: x = 0.2, 0.4 Hide answer | Show step-by-step
• Equation: 3y^2 - 2.55y - 1.0 = 0.5 Show answer | Show step-by-step Answer: y = -0.4, 1.25 Hide answer | Show step-by-step
• Equation: j^2 - 3.5j + 1.5 = 0 Show answer | Show step-by-step Answer: j = 0.5, 3 Hide answer | Show step-by-step
• Equation: c^2 + 3.75c - 1 = 0 Show answer | Show step-by-step Answer: c = -4, 0.25 Hide answer | Show step-by-step
• Equation: w^2 + 10.1w + 1 = 0 Show answer | Show step-by-step Answer: w = -10, -0.1 Hide answer | Show step-by-step
• Equation: x^2 -11.5x - 6 = 0 Show answer | Show step-by-step Answer: x = -0.5, 12 Hide answer | Show step-by-step

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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 ?

1. Answer :

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 

2. Answer :

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

3. Answer :

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

4. Answer :

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.

5. Answer :

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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Completing the square - interactive worksheet

Subject: Mathematics

Age range: 14 - 18

Resource type: Worksheet/Activity

Last updated

15 January 2012

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50 Completing The Square Practice Worksheet In 2020 With Images Practices Worksheets This

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Completing The Square Practice Worksheet

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