**Solving A Quadratic Equation By Completing The Square**

Every quadratic equation can be converted in the form:

(x + a)^{2} – b^{2} = 0 or (x – a)^{2} – b^{2} = 0.

**Steps:**

1. Bring, if required, all the term of the quadratic equation to the left hand side.

2. Express the terms containing x as x^{2} + 2xy or x^{2} – 2xy.

3. Add and subtract y2 to get x^{2} + 2xy + y^{2} – y^{2} or x^{2} – 2xy + y^{2} – y^{2}; which gives

(x + y)^{2} – y^{2} or (x – y)^{2} – y^{2}.

Thus,

(i) x^{2} + 8x = 0 ⇒ x^{2} + 2x × 4 = 0

⇒ x^{2} + 2x × 4 + 4^{2} – 4^{2} = 0

⇒ (x + 4)^{2} – 16 = 0

(ii) x^{2} – 8x = 0 ⇒ x^{2} – 2 × x × 4 = 0

⇒ x^{2} – 2 × x × 4 + 4^{2} – 4^{2} = 0

⇒ (x – 4)^{2} – 16 = 0

**Solving A Quadratic Equation By Completing The Square With Examples**

**Example 1: **Find the roots of the quadratic equation 2x^{2} – 7x + 3 = 0

(if they exist) by the method of completing the square.

**Sol. ** 2x^{2} – 7x + 3 = 0

** **[Dividing each term by 2]

**Example 2: **Find the roots of the quadratic equation 4x^{2} + 4√3x + 3 = 0

**Sol. ** 4x^{2} + 4√3x + 3 = 0

**Example 3: ** Find the roots of the quadratic equation 2x^{2} + x + 4 = 0

**Sol. **2x^{2} + x + 4 = 0

This is not possible as the square of a real number can not be negative.

**Foe More Solved Examples**