The solution of the rational equation is x = 7 or x = -4
<h3>How to solve for x in the rational equation?</h3>
Since we are are given the rational equation x/(x + 4) = 20/(x² + 3x - 4) + 2/(x - 1) we are required to solve for x.
So, factorizing out the denominator in the second term, 20/(x² + 3x - 4), we have
x/(x + 4) = 20/(x² + 3x - 4) + 2/(x - 1)
x/(x + 4) = 20/(x² + 4x - x - 4) + 2/(x - 1)
x/(x + 4) = 20/[x(x + 4) - (x + 4)] + 2/(x - 1)
x/(x + 4) = 20/[(x - 1)(x + 4)] + 2/(x - 1)
Now, since the L.C.M of the denominators is (x - 1)(x + 4), multiplying through by the L.C.M, we have
(x - 1)(x + 4)x/(x + 4) = (x - 1)(x + 4)20/[(x - 1)(x + 4)] + (x - 1)(x + 4)2/(x - 1)
(x - 1)x = 20 + (x + 4)2
Expanding the brackets, we have
x² - x = 20 + 2x + 8
Collecting like terms,we have
x² - x - 2x = 20 + 8
x² - 3x = 28
x² - 3x - 28 = 0
Factorizing,we have
x² - 7x + 4x - 28 = 0
x(x - 7) + 4(x - 7) = 0
(x - 7)(x + 4) =0
x - 7 = 0 or x + 4 = 0
x = 7 or x = -4
So, the solution of the rational equation is x = 7 or x = -4
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