Question

Find the roots of the following equation by Factorisation
x+1/x-1 x-1/x+1=5/6.,x#1,-1​

Answer 1

Step-by-step explanation:

(x+1)/(x-1)-(x-1)/(x+1) = 5/6.

So (x+1)(x+1)/[(x+1)(x-1)]-(x-1)(x-1)/[(x+1)(x-1)] = 5/6.

So [(x+1)(x+1)-(x-1)(x-1)]/[(x+1)((x-1)] = 5/6.

So [(x²+2x+1)-(x²–2x+1)]/(x²–1) = 5/6.

So (x²+2x+1-x+2x-1)/(x²–1) = 5/6.

So 4x/(x²–1) = 5/6.

So 4x = [(5/6)*(x–1) = (5/6)x–5/6.

So (5/6)x²–4x-5/6 = 0.

So 5x²–24x-5 = 0.

This is a quadratic equation in x, with a = 5, b = -24, c = -5.

Employing the quadratic formula;

x = [-b+/-√(b²–4ac)]/(2a)

= {-(-24)+/-√[(-24)²–4*5*(-5)]}/(2*5)

= {24+/-√[576-(-100)]}/100

= (24+/-√676)/10

= (24+/-26))/10

= 50/10 or -2/10

= 5 or -1/5 = -0.2 .

So x = 5

Or x = -1/5 = -0.2 .

CHECK:

(i) If x =5:

Then (x+1)/(x-1)-(x-1)/(x+1)

= (5+1)/(5–1)-(5–1)/(5+1)

= 6/4–4/6

= 18/12-8/12

= (18–8)/12

= 10/12

= 5/6. ✓

(ii) If x = -1/5:

Then (x+1)/(x-1)-(x-1)/(x+1)

= (-1/5+1)/(-1/5–1)-(-1/5–1)/(-1/5+1)

= (4/5)/(-6/5)-(-6/5)/(4/5)

= (4/5)*(-5/6)-(-6/5)*(5/4)

= -(4*5)/(5*6)-[-(6*5)/(5*4)]

= -4/6-(-6/4)

= -4/6+6/4

= 6/4-4/6

= 18/12–8/12

= (18–8)/12

= 10/12

= 5/6. ✓