Maths

86 MATHEMATICS
Example 14 : Find the roots of the following equations:
(i)
Solution :
x
1
3, x 0
(ii) 1 1 3, x 0, ✄ ✁ 2 ✂
x x x 2
✂ ✁
✄
(i)
1
x 3 ☎ ✆ . Multiplying throughout by x, we get
x
x 2 + 1 = 3x
i.e.,
x 2 – 3x + 1 = 0, which is a quadratic equation.
Here, a = 1, b = – 3, c = 1
So, b 2 – 4ac = 9 – 4 = 5 > 0
Therefore, x = 3 5 ✝
2
(Why?)
So, the roots are 3 5 and
3 5
✞
✟
.
2 2
(ii) 1 1 3, x
✄ 0, ✁ 2 ✂ .
x x 2
✄
As x ✠ 0, 2, multiplying the equation by x (x – 2), we get
(x – 2) – x =3x (x – 2)
=3x 2 – 6x
So, the given equation reduces to 3x 2 – 6x + 2 = 0, which is a quadratic equation.
Here, a = 3, b = – 6, c = 2. So, b 2 – 4ac = 36 – 24 = 12 > 0
Therefore, x = 6 12 6 2 3 3 3 ✡ ✡
✡
☛ .
6 6 3
☛
So, the roots are 3 3 and
3 3
✞
✟
.
3 3