Question

Solve the following equations:
$(x+9)(x-3)(x-7)(x+5)=385$.

Answer 1

$(x+9)(x-3)(x-7)(x+5)=385\left\{\right(x+9\left)\right(x-7\left)\right\}\left\{\right(x+5\left)\right(x-3\left)\right\}=385(x^2+9x-7x-63)(x^2+5x-3x-15)=385(x^2+2x-63)(x^2+2x-15)=385$

Put $x^2+2x=t$

$(t-63)(t-15)=385t^2-15t-63t+945=385t^2-78t+560=0t^2-8t-70t+560=0t(t-8)-70(t-8)=0(t-70)(t-8)=0t=70,8x^2+2x=70x^2+2x-70=\mathrm{0.........}\left(i\right)x^2+2x=8x^2+2x-8=\mathrm{0........}\left(ii\right)$

Solving $\left(i\right)$

$x^2+2x-70=0$

Using quadratic formula

$x=\frac{-2\pm \sqrt{4-4\left(1\right)(-70)}}{2}=\frac{-2\pm \sqrt{284}}{2}x=\frac{-2\pm 2\sqrt{71}}{2}=-1\pm \sqrt{71}$

Solving $\left(ii\right)$

$x^2+2x-8=0x^2+4x-2x-8=0x(x+4)-2(x+4)=0(x-2)(x+4)=0x=2,-4$

$\therefore$ The values of $x$ are $2,-4,-1+\sqrt{71}$ and $-1-\sqrt{71}$