NCERT Class 10 Mathematics

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.

In brief, this can be shown as follows:

x^2 + 4x =

442242

4

22 2

✆xx✂ ✁✝ ✄✆ ✁✝ ☎✆ ✂ ✁✝ ✄

✞ ✟ ✞ ✟ ✞ ✟

So, x^2 + 4x – 5 = 0 can be rewritten as

4 2

45

2

✠✌x☛ ✡✍ ☞ ☞

✎ ✏

=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

9 x^2 – 15x + 6 = 0

Now, 9 x^2 – 15x + 6 =^2

5

(3 ) 2 3 6

2

xx✑ ✒ ✒ ✓

=

22

(3 )^223 55 5 6

22 2

xx✖ ✗ ✗ ✘✔✙ ✕✚ ✖✔✙ ✕✚ ✘

✛ ✜ ✛ ✜

=

522 5

36

24

✠✌ x☞ ✡✍ ☞ ☛

✎ ✏

=

512

3

24

✠✌ x☞ ✡✍ ☞

✎ ✏