SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE EXAMPLES

Example 1 :

Solve the following quadratic equation by completing the square method.

x2 - 14x + 3  =  -10

Solution :

Write the given quadratic equation in the form :

ax2 + bx + c  =  0

Then, 

x2 - 14x + 3  =  -10

Add 10 to each side.

x2 - 14x + 13  =  0

Step 1 :

In the quadratic equation x2 - 14x + 13 = 0, the coefficient of x2 is 1. 

So, we have nothing to do in this step. 

Step 2 :

Subtract 13 from each side of the equation in step 1.

x2 - 14x  =  -13

Step 3 :

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

Then, 

x2 - 2(x)(7)  =  -13

Step 4 :

Now add 72 to each side to complete the square on the left side of the equation.  

Then, 

x2 - 2(x)(7) + 72  =  -13 + 72

(x - 7)2  =  -13 + 49

(x - 7)2  =  36

Take square root on both sides. 

(x - 7)2  =  √36

x - 7  =  ±6

x - 7  =  -6  or  x - 7  =  6

x  =  1  or  x  =  13

So, the solution is {1, 13}. 

Example 2 :

Solve the following quadratic equation by completing the square method.

x2 - 6x + 9  =  25

Solution :

Write the given quadratic equation in the form :

ax2 + bx + c  =  0

Then, 

x2 - 6x + 9  =  25

Subtract 25 from each side.

x2 - 6x - 16  =  0

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 in step 1.

x2 - 6x  =  16

Step 3 :

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

Then, 

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)(6) + 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}. 

Example 3 :

Solve the following quadratic equation by completing the square method.

x2 - 14x + 49  =  20

Solution :

Write the given quadratic equation in the form :

ax2 + bx + c  =  0

Then, 

x2 - 14x + 49  =  20

Subtract 20 from each side.

x2 - 14x + 29  =  0

Step 1 :

In the quadratic equation x2 - 14x + 29 = 0, the coefficient of x2 is 1. 

So, we have nothing to do in this step. 

Step 2 :

Subtract 29 from each side of the equation in step 1.

x2 - 14x  =  -29

Step 3 :

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

Then, 

x2 - 2(x)(7)  =  -29

Step 4 :

Now add 72 to each side to complete the square on the left side of the equation.  

Then, 

x2 - 2(x)(7) + 72  =  -29 + 72

(x - 7)2  =  -29 + 49

(x - 7)2  =  20

Take square root on both sides. 

(x - 7)2  =  √20

x - 7  =  ±25

x - 7  =  -25  or  x - 7  =  25

x  =  7 - 2√5  or  x  =  7 + 2√5

So, the solution is {7 - 2√5, 7 + 2√5}. 

Example 4 :

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

Add 2/3 to each side. 

x  =  2/3

So, the solution is 2/3. 

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