Question

Solve for $x:$ $\frac{1}{x+1}+\frac{2}{x+2}=\frac{4}{x+4},x\ne -1,-2,-4$

Answer 1

By cross multiplying, we get

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

$(3x+4)(x+4)=4(x^2+3x+2)$

$3x^2+12x+4x+16=4x^2+12x+8$

$x^2-4x-8=0$

We have a formula for solving quadratic equation $a{x}^2+bx+c=0$ which is

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

$x=\frac{4\pm \sqrt{16+32}}{2}=\frac{4\pm \sqrt{48}}{2}$

$x=\frac{4\pm 4\sqrt{3}}{2}=2\pm 2\sqrt{3}$

$x=2+2\sqrt{3},2-2\sqrt{3}$

$x=2+2\times 1.732,2-2\times 1.732$

$x=5.46,-1.46$