(Q10) Find the roots of the following equations

(i)x +
1
x
 = 3, x≠0   (ii)
1
x
 -
1
x - 2
 = 3, x≠0,2

Answer

(i)x +
1
x
 = 3. Multiplying both sides of equation by x, we get x2 + 1 = 3x

i.e x2 - 3x + 1 = 0,which is a quadratic equation.

Here, a = 1, b = -3, c = 1

So, b2 - 4ac = - = >0

Therefore, x=
3 ±√5
2
So,these roots are
3 +√
2
 and
3 -√
2
 .
(ii)
1
x
 -
1
x-2
 = 3, x≠0,2

As x≠0,2, multiplying the equation by x(x - 2), we get
(x - 2) - x = 3x (x - 2)
= 3x2 - 6x

So, the given equation reduces to 3x2 - 6x + 2 = 0, which is a quadratic equation.

Here, a = 3, b = -6, c = 2. So, b2 - 4ac = - = >0

Therefore, x=
6 ±
6
 =
6 ±2√
6
 =
3±√
3

So, the roots are
3+√
3
 and
3-√
3
 .