# Quadratic Equations and Linear Inequalities, Solving Quadratic Equation by Quadratic Formula, Relation Between Roots and Coefficients of a Quadratic Equation

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## Solving Quadratic Equation by Quadratic Formula

Recall the solution of a standard quadratic equation by the “Method of Completing Squares”

Roots of the above quadratic equation are given by

and

Where is called the discriminant of the quadratic equation.

**Note:**

For a quadratic equation if

(i) , the equation will have two real and unequal roots

(ii) , the equation will have two real and equal roots and both roots are equal to

(iii) , the equation will have two conjugate complex (imaginary) roots.

Example:

For what values of the quadratic equation will have equal roots?

Solution:

The given quadratic equation is

Here,

For equal roots,

, which are the required values of

## Relation between Roots and Coefficients of a Quadratic Equation

You have learnt that, the roots of a quadratic equation are and

Let

And

Adding (1) and (2), we have

Sum of the roots

Product of the roots

(3) and (4) are the required relationships between roots and coefficients of a given quadratic equation. These relationships helps to find out a quadratic equation when two roots are given.

Example:

If, are the roots of the equation find the value of:

(a) (b)

Solution:

It is given that are the roots of the quadratic equation .

And

Now, [By (1) and (2)]

Now, [By (1) and (2)]