To solve quadratic equations using completing the square method, the given quadratic equation must be in the form of
ax^{2} + bx + c = 0
The following steps will be useful to solve a quadratic in the above form using completing the square method.
Step 1 :
In the given quadratic equation ax^{2} + bx + c = 0, divide the complete equation by a (coefficient of x^{2}).
If the coefficient of x^{2} is 1 (a = 1), the above process is not required.
Step 2 :
Move the number term (constant) to the right side of the equation.
Step 3 :
In the result of step 2, write the "x" term as a multiple of 2.
Examples :
6x should be written as 2(3)(x).
5x should be written as 2(x)(5/2).
Step 4 :
The result of step 3 will be in the form of
x^{2} + 2(x)y = k
Step 4 :
Now add y^{2} to each side to complete the square on the left side of the equation.
Then,
x^{2} + 2(x)y + y^{2} = k + y^{2}
Step 5 :
In the result of step 4, if we use the algebraic identity
(a + b)^{2} = a^{2} + 2ab + b^{2}
on the left side of the equation, we get
(x + y)^{2} = k + y^{2}
Step 6 :
Solve (x + y)^{2} = k + y^{2 }for x by taking square root on both sides.
Example 1 :
Solve the following quadratic equation by completing the square method.
9x^{2} - 12x + 4 = 0
Solution :
Step 1 :
In the given quadratic equation 9x^{2} - 12x + 4 = 0, divide the complete equation by 9 (coefficient of x^{2}).
x^{2} - (12/9)x + (4/9) = 0
x^{2} - (4/3)x + (4/9) = 0
Step 2 :
Subtract 4/9 from each side.
x^{2} - (4/3)x = - 4/9
Step 3 :
In the result of step 2, write the "x" term as a multiple of 2.
Then,
x^{2} - (4/3)x = - 4/9
x^{2} - 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,
x^{2} - 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.
Example 2 :
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 :
ax^{2} + 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 = 3x^{2} - 3x + 2x - 2
5x + 7 = 3x^{2} - x - 2
0 = 3x^{2} - 6x - 9
or
3x^{2} - 6x - 9 = 0
Divide the entire equation by 3.
x^{2} - 2x - 3 = 0
Step 1 :
In the quadratic equation x^{2} - 2x - 3 = 0, the coefficient of x^{2} is 1.
So, we have nothing to do in this step.
Step 2 :
Add 3 to each side of the equation x^{2} - 2x - 3 = 0.
x^{2} - 2x = 3
Step 3 :
In the result of step 2, write the "x" term as a multiple of 2.
Then,
x^{2} - 2x = 3
x^{2} - 2(x)(1) = 3
Step 4 :
Now add 1^{2} to each side to complete the square on the left side of the equation.
Then,
x^{2} - 2(x)(1) + 1^{2} = 3 + 1^{2}
(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}.
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