A formula to calculate the roots or solutions of a quadratic
equation ax^{2} + bx + c = 0 is known as the **Quadratic Formula** which is given as,

where a is the coefficient of x^{2}, 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 b^{2} > 4ac, there are two distinct real roots.

When b^{2} = 4ac, there is a single real root.

When b^{2} < 4ac, there is no real roots.

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**Derivation of
Quadratic Formula**

Consider the general quadratic equation

ax^{2} + bx + c = 0, where a, b and c are constants and
a≠ 0.

Rewriting the given equation as

ax^{2} + bx = – c

Dividing both sides by the coefficient of x^{2} i.e. a we get

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

Hence the Quadratic Formula is derived.

**Workout Examples**

Example 1: Solve x^{2} – 7x + 12 = 0 by using quadratic
formula.

Solution: Here,

x^{2} – 7x + 12 = 0

Comparing this equation with ax^{2} + bx + c = 0 we get,

a = 1

b = –7

c = 12

Now, by using quadratic formula,

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

Solution: Here,

(p – q)x^{2} + (p + q)x + 2q = 0

Comparing this equation with ax^{2} + bx + c = 0 we get,

a = (p – q)

b = (p + q)

c = 2q

Using quadratic formula,

or, 6x^{2}-18x+12 = 3x^{2}-5x

or, 6x^{2}–18x+12–3x^{2}+5x = 0

or, 3x^{2}–13x+12= 0

Comparing this equation with ax^{2} + 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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