Question

Solve for x:
1/(x+1) + 2/(x+2) = 4/(x+4) , x ≠ -1, -2, -3

Answer 1

Answer:

x= (2±√3) is the value for

Step-by-step explanation:

$\frac{1}{x+1} + \frac{2}{x+2} = \frac{4}{x+4}$

$\frac{(x+2)+2 (x+1)}{(x+2)(x+1)} =\frac{4}{x+4}$

$\frac{(x+2) + (2x+2)}{(x^{2 } +2x+x+2} =\frac{4}{(x+4)}$

$\frac{3x+4}{x^{2}+3x+ 2} =\frac{4}{x+4}$

$(3x +4 ) (x+4) =4 (x^{2} + 3x +2 )\\3x^{2} + 12x+4x + 16 = 4x^{2} + 12x+ 8\\4x^{2} -3x^{2} +12x-12x -4x +8- 16 =0$

$x^{2} - 4x + 8 = 0$

As we can see that the equation cannot be factorize using rational number. Hence the values of the roots are:

$x= \frac{-b \sqrt{b^{2}-4ac } }{2a}$

The value of a = 1, b = -4 and c = -8

$x= \frac{-(-4) +-(\sqrt{16-4(1) (-8)}}{2*1} \\x=\frac{4+- \sqrt{16+32} }{2} \\x=\frac{4 +- \sqrt{48} }{2} \\x=\frac{4 +- \sqrt{16 *3} }{2} \\x=\frac{4 +- 4\sqrt{3} }{2} \\x= \frac{4(1+- \sqrt{3} }{2} \\x=2(1+-\sqrt{3})$