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