In this section, we will learn about quadratic equations, their concepts, roots, and graphs.

Equation: It is an equation of the form  ax^{2}+bx+c= 0 . Where a, b, c are real numbers , a ≠ 0 and x is the variable.
We can also say that it is a polynomial equation of degree 2 with a single variable.

Standard form of a quadratic equation.
ax^{2}+bx+c= 0 and a ≠ 0

Examples:

1. x^{2}+2x+2= 0
Here a= 1, b = 2 and c=2.
2. x^{2}+3x-1= 0
Here a= 1, b = 3 and c = – 1.
3. 4x^{2}-8x= 0
Here a= 4, b = -8 and c = 0.
4. 2x^{2}-1= 0
Here a= 2, b = 0 and c = -1.

Question: Check whether the following equations are quadratic or not.
i) x^{2}+5x+6= 0
Answer: Yes, the above equation is a quadratic equation. It is of the form ax^{2}+bx+c= 0 where a= 1, b = 5 and c = 6.
ii)x^{2}+2= 0
Answer: Yes, the above equation is a quadratic equation. It is of the form ax^{2}+bx+c= 0 where a= 1, b = 0 and c = 2.
iii) x-2= 0
Answer: No, the above equation is not a quadratic equation because it is not of the form ax^{2}+bx+c= 0. As here a = 0. For a quadratic equation  a ≠ 0.

### Methods of finding the roots of quadratic equations

When we put x = α in the quadratic equation of the form ax^{2}+bx+c= 0 ,  such that aα ^{2}+bα +c= 0 i.e α  satisfies the equation. Thus, α is called the root or solution of quadratic equation.

The roots of quadratic equation ax^{2}+bx+c= 0 , a 0 can be given by,

### x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}

provided b^{2}-4ac ≥ 0.

Question : Find the roots of the quadratic equation 3x^{2}- 5x + 2= 0 by quadratic formula.
Solution: Given, 3x^{2}- 5x + 2= 0
Here a = 3 , b = -5 and c= 2

x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}

Now b^{2}-4ac = (-5)^{2} – 4.2.3 = 25 – 24 = 1

x=\frac{-(-5)\pm \sqrt{1}}{2.3}

⇒ x=\frac{5\pm 1}{6}

⇒ x=1 or x=\frac{2}{3}

Thus. the roots are 1 and \frac{2}{3}
2. #### Finding roots by Factorisation:

This can be done by splitting the middle term.
Let us understand this by an example.
i) x^{2}-3x-10= 0
This can written as
x^{2}-5x+2x-10= 0
x ( x – 5) + 2 ( x – 5) = 0
( x -5 ) (x + 2) = 0
Thus, x = – 2 and x = 5
Here we found the roots of the equation by factorising the equation into two linear factors and equating each factor to zero.

## Nature of roots and Discriminant

We know the roots of quadratic equation ax^{2}+bx+c= 0 , a 0 can be given by,

x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}

It is to be noted that b^{2}-4ac determines the nature of the roots. Also,  b^{2} – 4ac is called the discriminant of quadratic equation.

 S.No Discriminant Nature of roots 1 b^{2}-4ac> 0 Two distinct and real roots 2 b^{2}-4ac = 0 Two equal and real roots 3 b^{2}-4ac< 0 No real roots