Maths

78 MATHEMATICS
The process is as follows:
x 2 + 4x =(x 2 + 4 2 x ) + 4 2 x
=x 2 + 2x + 2x
= (x + 2) x + 2 × x
=(x + 2) x + 2 × x + 2 × 2 – 2 × 2
=(x + 2) x + (x + 2) × 2 – 2 × 2
=(x + 2) (x + 2) – 2 2
=(x + 2) 2 – 4
So, x 2 + 4x – 5 = (x + 2) 2 – 4 – 5 = (x + 2) 2 – 9
So, x 2 + 4x – 5 = 0 can be written as (x + 2) 2 – 9 = 0 by this process of completing
the square. This is known as the method of completing the square.
So,
In brief, this can be shown as follows:
x 2 + 4x =
2 2 2
✁ ✁ ✁
✂ ✄ ☎ ✂ ✄
✆ ✝ ✆ ✝ ✆ ✝
✞ ✟ ✞ ✟ ✞ ✟
x
4 4 4
x
2 2 2
4
x 2 + 4x – 5 = 0 can be rewritten as
2
✠ ✡
☛ ☞ ☞
✌ ✍
✎ ✏
4
x
2
4 5 =0
i.e., (x + 2) 2 – 9 = 0
Consider now the equation 3x 2 – 5x + 2 = 0. Note that the coefficient of x 2 is not
a perfect square. So, we multiply the equation throughout by 3 to get
9x 2 – 15x + 6 = 0
Now, 9x 2 – 15x + 6 =
=
2 5
✑ ✒ ✒ ✓
(3 x) 2 3x
2
6
2 2
2 5 ✔ ✕ ✔ ✕ 5 5
(3 x) 2 ✖ ✗ ✗ ✘ ✖ ✘
✙ ✚ ✙ ✚
3x
6
2 2 2
✛ ✜ ✛ ✜
=
2
5 ✠ ✡ 25
3x
☞ ☞ ☛
✌ ✍ 6
2 4
✎
✏
=
✠
✌ 3x
☞
✎
2
5✡
1
☞ ✍
2 4
✏