82 MATHEMATICS
Example 9 : Find the roots of 4x^2 + 3x + 5 = 0 by the method of completing the
square.
Solution : Note that 4x^2 + 3x + 5 = 0 is the same as
(2x)^2 + 2 × (2x) ×33 3^22
5
44 4
✂ ✁ ✄ ✁ ✂
☎ ✆ ☎ ✆
✝ ✞ ✝ ✞
=0
i.e.,
392
25
416
☎ x✂ ✁✆ ✄ ✂
✝ ✞=0
i.e.,
372 1
2
416
✟☛ x✡ ✠☞ ✡
✌ ✍=0
i.e.,
3 2
2
4
✟☛ x✡ ✠☞
✌ ✍=
71
0
6
✎
✏
But3 2
2
4
✑✔ x✓ ✒✕
✖ ✗cannot be negative for any real value of x (Why?). So, there isno real value of x satisfying the given equation. Therefore, the given equation has no
real roots.
Now, you have seen several examples of the use of the method of completing
the square. So, let us give this method in general.
Consider the quadratic equation ax^2 + bx + c = 0 (a ✘ 0). Dividing throughout bya, we get^20
xxbc
aa
✙ ✙ ✚
This is the same as
22
0
22
x bbc
aaa✑ ✓ ✒ ✛✑ ✒ ✓ ✜
✔ ✕ ✔ ✕
✖ ✗ ✖ ✗
i.e.,
(^22)
2
4
(^24)
x bbac
a a
✑ ✒ ✛
✔ ✓ ✕ ✛
✖ ✗
= 0
So, the roots of the given equation are the same as those of
(^22)
2
4
0,
2 4
bbac
x
a a✑ ✒ ✛
✔ ✓ ✕ ✛ ✜
✖ ✗
i.e., those of(^22)
2
4
2 4
bbac
x
a a