Solve each of the following quadratic equations by completing the square method.
Problem 1 :
9x2 - 12x + 4 = 0
Problem 2 :
x2 - 6x - 16 = 0
Problem 3 :
x2 - 5x + 6 = 0
Problem 4 :
(5x + 7)/(x - 1) = 3x + 2
Problem 1 :
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.
Problem 2 :
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 :
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 :
(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}.
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