**Solving Quadratic Equations by Completing the Square :**

In this section, you will learn how to solve quadratic equation using by completing the square.

To apply completing the square method, the quadratic equation must be in the form of

ax^{2} + bx + c = 0

**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}.

After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations by completing the square method.

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