Solution of a Quadratic Equation by Completing the Square
Following steps are involved in solving a quadratic equation by quadratic formula
· Consider the equation , where a ¹ 0
· Dividing throughout by ‘a’, we get
· Add and subtract , we get
,
· If taking square root of both sides, we obtain
Therefore
The Quadratic Formula: Quadratic equation, where a, b, c are real number and , has the roots as
Problems based on Solution of a Quadratic Equation by Completing the Square with Solutions
Problem:
Solve by completing the square :
Solution:
(i)
Þ
Adding i.e. on both sides, we have
Þ
or
Þ
Þ
So, the solutions are
Problem:
By using the method of completing the square, show that the equation has no real roots.
Solution:
Þ [Dividing the equation by 2]
Þ
Adding i.e. on both sides, we have
Þ
Þ
Þ
RHS is negative but cannot be negative for any real values of x.
[ square of any real number is non-negative]
Hence, the given equation has no real roots.
Problem:
Find the roots of the following equation
by the method of completing the square.
Solution:
We have,
Þ [Dividing the equation by 4]
Þ
Adding on both sides we have
Þ
Þ
Þ [taking square root of both sides]
Þ
Þ
Hence, the roots are and
Problem:
Solve the equation by the method of completing the square.
Solution:
We have,
Þ
Adding on both sides we have
Þ
Þ
Þ
Þ
Þ
or
Þ
Hence, the roots are and 1.
Problem:
Using quadratic formula, solve the following quadratic equation for x:
Solution:
Comparing the given equation
with the standard quadratic equation
We have a = 1
b = –2a
And
Now Discriminant
Þ Þ
Using the quadratic formula, we get
Hence or
Problem:
Using quadratic formula solve the following quadratic equations:
(i) (ii)
Solution:
(i) We have,
Comparing the equation with we have
and
\
[ square of any real number is non-negative]
So, the given equation has real roots given by
and,
Hence the roots are and -1
(ii) We have,
Comparing this equation with we have
and
\
Þ
Þ
Þ
Þ
Þ [ square of any real number is non-negative]
So, the roots of the given equation are real and are given by
and,
\ Solution is and
Problem:
Using the quadratic formula, solve the equation
Solution:
Comparing the given equation with
We have
Discriminant
[ square of any real number is non-negative]
Hence, the given equation has real roots given by
The roots are
and
or and
or and
Hence, are the required solutions.