# Solution of a Quadratic Equation by Completing the Square

Steps for completing the square

**Step 1**

Write an equivalent equation with only the x^{2 }term and the x term on the left side of the equation. The coefficient of the x^{2} term must be 1.

**Step 2**

Add the square of one-half the coefficient of the x term to both sides of the equation.

**Step 3**

Express the left side of the equation as a perfect square.

**Step 4**

Solve for x.

**Theorem**

For any quadratic equation in the form ax^{2} + bx + c = 0, where a â‰ 0, the two solutions are

**Proof **

We will prove the quadratic theorem by completing the square on ax^{2} + bx + c = 0.

ax^{2} + bx + c = 0

This completes the proof. Let us see what we have proved. If our equation is in the form ax^{2} + bx + c = 0 (standard form), where a â‰ 0, the two solutions are always given by the formula

This formula is known as the quadratic formula. If we substitute the coefficients a, b and c of any quadratic equation in standard form in the formula, we need to perform some basic arithmetic to arrive at the solution set.

**Note**

This method was developed by the noted mathematician Sridharcharya. If b

^{2 }- 4ac â‰¥ 0, then quadratic equation ax

^{2 }+ bx + c = 0 has two real roots Î± and Î² which will be given by,

Î± = and Î² =

**Discriminant**

The quantity 'b

^{2 }- 4ac' is called, discriminant of the equation ax

^{2 }+ bx + c = 0 and is denoted by the letter 'D'.

Therefore D = b

^{2 }- 4ac.