# Working with Quadratic Equations

## What is a Quadratic Equation?

A quadratic equation is an equation of the order two i.e. the maximum power to which the variable is raised is 2.

It is denoted as:

ax² + bx + c = 0

where a, the co-efficient of x² can be any integer except 0.

b, the co-efficient of x, could be any integer.

c, a constant could be any integer.

Some examples of quadratic equations are :

x² + 5x + 6 = 0

7m² + 19m - 32 = 0

4y² - 5y + 12 = 0

## Roots of a Quadratic Equation

The roots of a quadratic equation are the values of the variable (x, m and y in the above examples) for which the equation holds true.

In other words, if we have a quadratic equation such as x² + 5x + 6 = 0 , then the roots of the equation are those values of x for which x² + 5x + 6 will equal zero. In the above example, if we put x = –2, we get

(–2)² + 5(–2) + 6 = 4 – 10 + 6 = 0

Hence, –2 is a root of the equation.

Similarly, if we put x = –3, we get

(–3)² + 5(–3) + 6 = 9 – 15 + 6 = 0

Hence –3 is also a root of the equation.

Every quadratic equation has two roots only.

## How to find the roots of a Quadratic Equation

There are two methods for finding the roots of a Quadratic Equation.

### Method 1 : Factorization

Consider the equation x² + 5x + 6 = 0

If we can express the above equation in the form (x + ?)(x + ?) = 0, we can find the roots.

The process of factorization is shown below.

x² + 5x + 6 = 0

∴ x² + 3x + 2x + 6 = 0

∴ x(x + 3) + 2(x + 3) = 0

∴ (x + 2)(x + 3) = 0

As the product of (x + 2) and (x + 3) is zero, we can say,

Either (x + 2) = 0 OR (x + 3) = 0

∴ x = –2 OR x = –3

Therefore, –2 and –3 are the roots of the equation x² + 5x + 6 = 0

Generally speaking, if you have to factorize an equation of the nature x² + bx + c = 0 , look out for 2 numbers whose sum is equal to ‘b’ and product equal to ‘c’. Then change the sign of the 2 numbers. These values are the roots of the quadratic equation.

e.g. factorize x² – 6x + 8 = 0

Here b = –6, c = 8

We have to think of two numbers whose sum is –6 and product is 8.

The numbers are: –2 and –4. Hence the roots will be +2 and +4

[We can write the equation as: (x – 2)(x – 4) = 0, in which case the roots will be 2 & 4]

e.g. find the roots of the equation x² + x – 20 = 0

Here, b = 1, c = –20

Therefore, we have to think of two numbers whose sum is 1 and product –20

These numbers are +5 and –4

∴ The roots are –5 and +4 [since (x + 5)(x – 4) = 0]

### Method 2: Formula

There is a formula available for finding the roots of any quadratic equation of the form:

ax² + bx + c = 0

The roots of the above quadratic are given by

–b ± √(b² – 4ac)

2a

Let us apply this formula to the following example.

e.g. x² + 5x + 6 = 0

We have already found the roots of this equation using factorization method. Let us now apply the formula method.

Here, a = 1 , b = 5, c = 6

–5 ± √(5² – 4×1×6)

2×1

∴ = – 4/2 or – 6/2 = – 2 or – 3

### Some observations

*When b² – 4ac = 0, the roots will be equal. i.e. there will be only one root value*

*When b = 0, the roots will be equal in value but will have opposite sign.*

## Constructing a Quadratic Equation from its roots

If the roots of a quadratic equation are given, we can form the equation using the roots.

If *m* and *n* are the roots of a quadratic equation, then the equation will be

* **x² – (m + n)x + (m×n) = 0*

e.g. The roots of a quadratic equation are 3 and 5. What is the equation ?

Here, m = 3 and n = 5.

The equation will be x² – (m + n)x + (m×n) = 0

∴ x² – (3 + 5)x + (3 × 5) = 0

∴ x² – 8x + 15 = 0

## Some standard results for Quadratic Equations

If *m* and *n* are the roots of a quadratic equation *ax² + bx + c = 0* , then

The sum of roots of a quadratic equation,

m + n = – b / a

Product of roots of a quadratic equation,

m × n = c / a

If a = c, then the roots will be the reciprocal of each other