Quadratic Formula

 

A formula to calculate the roots or solutions of a quadratic equation ax² + bx + c = 0 is known as the Quadratic Formula which is given as,

where a is the coefficient of x², b is the coefficient of x and c is the constant term. We use this formula to find the required solutions (roots) of the given quadratic equation.

Some special conditions:

When b² > 4ac, there are two distinct real roots.

When b² = 4ac, there is a single real root.

When b² < 4ac, there is no real roots.

 

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Derivation of Quadratic Formula

 

Consider the general quadratic equation

ax² + bx + c = 0, where a, b and c are constants and a≠ 0.

Rewriting the given equation as

ax² + bx = – c

Dividing both sides by the coefficient of x² i.e. a we get

In order to make the left-hand side a perfect square, we take half the coefficient of x,

Taking square root to both sides, we get

Hence the Quadratic Formula is derived.

 

Workout Examples

 

Example 1: Solve x² – 7x + 12 = 0 by using quadratic formula.

Solution: Here,

x² – 7x + 12 = 0

Comparing this equation with ax² + bx + c = 0 we get,

a = 1

b = –7

c = 12

Now, by using quadratic formula,

 

Example 2: Solve (p – q)x² + (p + q)x + 2q = 0 by using quadratic formula.

Solution: Here,

(p – q)x² + (p + q)x + 2q = 0

Comparing this equation with ax² + bx + c = 0 we get,

a = (p – q)

b = (p + q)

c = 2q

Using quadratic formula,

 

or, 6x²-18x+12 = 3x²-5x

or, 6x²–18x+12–3x²+5x = 0

or, 3x²–13x+12= 0

Comparing this equation with ax² + bx + c = 0 we get,

a = 3

b = –13

c = 12

Now, by using quadratic formula,

 

If you have any questions or problems regarding the Quadratic Formula, you can ask here, in the comment section below.

 

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