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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) ]