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Quadratic equation

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Quadratic equation [#permalink] New post 01 Jan 2007, 19:58
Quick question on solving the following equation..

***********
1/x + 1/(x - 4) = 5/24.

The explanation given in the answer is --
The solution of this equation is = 12 (only the +ve root counts here)

*************

Can someone help me with a way to find out the roots ?
I went through the laborous task of

(x+x+4) / (x*(x+4)) = 5/24 and then using the formula to find out the root.

There should be an easier way to find this.

Thanks...
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 [#permalink] New post 01 Jan 2007, 23:14
1/x + 1/(x - 4) = 5/24
<=> (x-4 + x)/(x*(x-4)) = 5/24

=> 2*x - 4 = 5/24(x^2 - 4*x) with x != 0 and x != 4
<=> 5*x^2 - 20*x - 24*(2*x-4) = 0
<=> 5*x^2 - 20*x - 48*x + 96 = 0
<=> 5*x^2 - 68*x + 96 = 0

Once arrived, it's better to use the reduced Delta than to use the usual Delta.

reduced Delta = (b/2)^2 - a*c = 34^2 - 5*96 = 676 = 26^2

x1 = (-(b/2) + sqrt(reduced Delta) ) / a = (34 + 26) / 5 = 12

x2 = (-(b/2) - sqrt(reduced Delta) ) / a = (34 - 26) / 5 = 8/5

Finally, as x1 and x2 are neither eqaul to 0 nor to 4, we have x = 12 and x = 8/5 like solutions for the original equation.

To be sure,
o 1/12 + 1/(12-4) = 1/12 + 1/8 = (2+3)/(12*2) = 5/24
o 5/8 + 1/(8/5 - 4) = 5/8 + 5/(8-20) = 5/8 - 5/12 = (15 - 10)/24 = 5/24
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 [#permalink] New post 03 Jan 2007, 19:52
I sugget entering values from answer choice for such question. It may work.
Both the methods given in previous posts are time consuming.
  [#permalink] 03 Jan 2007, 19:52
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