Question

find the roots of the following equation by factorisation X+1/x-1 - x-1/X+1=5/6;Xi's not equal to 1,-1​

Answer 1

Answer:

(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. ✓

Step-by-step explanation:

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