# SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE WORKSHEET

Solving Quadratic Equations by Completing the Square Worksheet :

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

Before look at the worksheet, if you would like to learn how to solve quadratic equations by completing the square,

## Solving Quadratic Equations by Completing the Square Worksheet - Problems

Problem 1 :

Solve the following quadratic equation by completing the square method.

9x2 - 12x + 4  =  0

Problem 2 :

Solve the following quadratic equation by completing the square method.

x2 - 6x - 16  =  0

Problem 3 :

Solve the following quadratic equation by completing the square method.

x2 - 5x + 6  =  0

Problem 4 :

Solve the following quadratic equation by completing the square method.

(5x + 7)/(x - 1)  =  3x + 2 ## Solving Quadratic Equations by Completing the Square Worksheet - Solutions

Problem 1 :

Solve the following quadratic equation by completing the square method.

9x2 - 12x + 4  =  0

Solution :

Step 1 :

In the given quadratic equation 9x2 - 12x + 4 = 0, divide the complete equation by 9 (coefficient of x2).

x2 - (12/9)x + (4/9)  =  0

x2 - (4/3)x + (4/9)  =  0

Step 2 :

Subtract 4/9 from each side.

x2 - (4/3)x  =  - 4/9

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 - (4/3)x  =  - 4/9

x2 - 2(x)(2/3)  =  - 4/9

Step 4 :

Now add (2/3)2 to each side to complete the square on the left side of the equation.

Then,

x2 - 2(x)(2/3) + (2/3)2  =  - 4/9 + (2/3)2

(x - 2/3)2  =  - 4/9 + 4/9

(x - 2/3)2  =  0

Take square root on both sides.

(x - 2/3)2  =  0

x - 2/3  =  0

x  =  2/3

So, the solution is 2/3.

Problem 2 :

Solve the following quadratic equation by completing the square method.

x2 - 6x - 16  =  0

Solution :

Step 1 :

In the quadratic equation x2 - 6x - 16 = 0, the coefficient of x2 is 1.

So, we have nothing to do in this step.

Step 2 :

Add 16 to each side of the equation x2 - 6x - 16  =  0.

x2 - 6x  =  16

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 - 6x  =  16

x2 - 2(x)(3)  =  16

Step 4 :

Now add 32 to each side to complete the square on the left side of the equation.

Then,

x2 - 2(x)(3) + 32  =  16 + 32

(x - 3)2  =  16 + 9

(x - 3)2  =  25

Take square root on both sides.

(x - 3)2  =  √25

x - 3  =  ±5

x - 3  =  -5  or  x - 3  =  5

x  =  -2  or  x  =  8

So, the solution is {-2, 8}.

Problem 3 :

Solve the following quadratic equation by completing the square method.

x2 - 5x + 6  =  0

Solution :

Step 1 :

In the quadratic equation x2 - 5x + 6 = 0, the coefficient of x2 is 1.

So, we have nothing to do in this step.

Step 2 :

Subtract 6 from each side of the equation x2 - 5x + 6 = 0.

x2 - 5x  =  -6

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 - 5x  =  -6

x2 - 2(x)(5/2)  =  -6

Step 4 :

Now add (5/2)2 to each side to complete the square on the left side of the equation.

Then,

x2 - 2(x)(5/2) + (5/2)2  =  -6 + (5/2)2

(x - 5/2)2  =  -6 + 25/4

(x - 5/2)2  =  -24/4 + 25/4

(x - 5/2)2  =  (-24 + 25)/4

(x - 5/2)2  =  1/4

Take square root on both sides.

(x - 5/2)2  =  √(1/4)

x - 5/2  =  ±1/2

x - 5/2  =  -1/2  or  x - 5/2  =  1/2

x  =  -1/2 + 5/2  or  x  =  1/2 + 5/2

x  =  4/2  or  x  =  6/2

x  =  2  or  x  =  3

So, the solution is {2, 3}.

Problem 4 :

Solve the following quadratic equation by completing the square method.

(5x + 7)/(x - 1)  =  3x + 2

Solution :

Write the given quadratic equation in the form :

ax2 + bx + c  =  0

Then,

(5x + 7)/(x - 1)  =  3x + 2

Multiply each side by (x - 1).

5x + 7  =  (3x + 2)(x - 1)

Simplify.

5x + 7  =  3x2 - 3x + 2x - 2

5x + 7  =  3x2 - x - 2

0  =  3x2 - 6x - 9

or

3x2 - 6x - 9  =  0

Divide the entire equation by 3.

x2 - 2x - 3  =  0

Step 1 :

In the quadratic equation x2 - 2x - 3 = 0, the coefficient of x2 is 1.

So, we have nothing to do in this step.

Step 2 :

Add 3 to each side of the equation x2 - 2x - 3 = 0.

x2 - 2x  =  3

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 - 2x  =  3

x2 - 2(x)(1)  =  3

Step 4 :

Now add 12 to each side to complete the square on the left side of the equation.

Then,

x2 - 2(x)(1) + 12  =  3 + 12

(x - 1)2  =  3 + 1

(x - 1)2  =  4

Take square root on both sides.

(x - 1)2  =  4

x - 1  =  ±2

x - 1  =  -2  or  x - 1  =  2

x  =  -1  or  x  =  3

So, the solution is {-1, 3}. After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations by completing the square method.

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