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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