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 = ±2√5
x - 7 = -2√5 or x - 7 = 2√5
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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