Question

X²+6x+24=10*\sqrt{ x^{2} +6x}

Answer 1

$x^{2} +6x \geq 0 \\ x^{2} +6x = 0 \\ x(x+6)=0 \\ x=0;x=-6$
x∈(-oo;-6] U [0;+oo)
$x^{2} +6x=t \\ x^{2} +6x+24=10*\sqrt{ x^{2} +6x} \\ t+24=10 \sqrt{t} \\ (t+24) ^{2} =(10 \sqrt{t}) ^{2} \\t ^{2} +48t+576=100t\\ t ^{2} -52t+576=0 \\ D=52 ^{2} -4*576=2704-2304=400=20^{2} \\ t= \frac{52-20}{2} =16 \\ t= \frac{52+20}{2} =36 \\ t=x^{2} +6x \\ x^{2} +6x=16 \\ x^{2} +6x-16 =0\\ x=-8 \\ x=2 \\ x^{2} +6x=36 \\ x^{2} +6x-36=0\\ D=36-4*(-36)=36+144=180=(6 \sqrt{5} ) ^{2} \\ x= \frac{-6+6 \sqrt{5} }{2} =-3+3 \sqrt{5} \\ x=-3-3 \sqrt{5}$