
Completing the Square Method
In completing the square
method for solving a
quadratic equation of the form ax2 + bx + c = 0, we transpose the
constant term c to the right hand side of the equation and then the left hand
side is expressed as a perfect square expression.
We use the following steps for solving quadratic
equation of the form ax2 + bx + c = 0 by completing the square
method:
Steps:
1. Rewrite the equation so that the terms
containing x2 and x (i.e. variable) are on the left side and the
constant term is on the right side.
2. If the coefficient of x2 is
other than 1, then reduce the coefficient of x2 as 1 by dividing on
both sides by the coefficient of x2.
3. Add the square of half of the coefficient
of x on both sides of the equation to make the left hand side of the equation a
complete square.
4. Express the left hand side of the equation
as a perfect square of a binomial.
5. Use the square root property on both sides
and calculate the roots.
This process of Completing the Square Method
will be clear by the following worked out examples.
Workout Examples
Example 1: Solve the following quadratic
equations by completing the square method.
a.
x2 – 2x – 8 =0
b.
2x2 + 5x – 12 = 0
c.
3x2 – 5x + 2 = 0
Solution: a) Here,
x2 – 2x – 8 =
0
or, x2
– 2x = 8
or, x2
– 2x + 1 = 8 + 1 [adding both sides by 1]
or, (x
– 1)2 = 9
or, x
– 1 = ±√9
or, x
– 1 = ±3
or, x
= 1 ± 3
Taking +ve
sign,
x
= 1 + 3 = 4
Taking –ve sign,
x
= 1 – 3 = –2
∴ x = 4, –2
Solution: b) Here,
2x2 + 5x
– 12 = 0
or, 2x2 + 5x = 12
or, (2x2 + 5x)/2 = 12/2 [dividing both sides by 2]
or, x2 +5x/2 = 6
or, x2 + 2.x.5/4 + (5/4)2
= 6 + (5/4)2 [adding both sides by (5/4)2]
or, (x + 5/4)2 = 6 + 25/16
or, (x + 5/4)2 = 121/16
or, x + 5/4 = ±√(121/16)
or, x + 5/4 = ± 11/4
or, x = – 5/4 ± 11/4
Taking +ve sign,
x
= – 5/4 + 11/4 = (–5 + 11)/4 = 6/4 = 3/2
Taking –ve sign,
x
= – 5/4 – 11/4 = (–5 – 11)/4 = –16/4 = –4
∴ x = 3/2, –4
Solution: c) Here,
3x2 – 5x
+ 2 = 0
or, 3x2 – 5x = –2
or, (3x2 – 5x)/3 = –2/3 [dividing both sides by 3]
or, x2 –5x/3 = –2/3
or, x2 – 2.x.5/6 + (5/6)2 = –2/3 + (5/6)2
[adding
both sides by (5/6)2]
or, (x – 5/6)2 = –2/3 + 25/36
or, (x – 5/6)2 = (–24 + 25)/36
or, x – 5/6 = ±√(1/36)
or, x – 5/6 = ± 1/6
or, x = 5/6 ± 1/6
Taking +ve sign,
x
= 5/6 + 1/6
= (5 + 1)/6 = 6/6 = 1
Taking –ve sign,
x
= 5/6 – 1/6 = (5 –1)/6 = 4/6 = 2/3
∴ x = 1, 2/3
You can comment your
questions or problems on solving quadratic equations by ‘completing the square
method’ here.
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