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

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) ]