Question

$( x^{2} +14x+24)( x^{2} +11x+24)=4 x^{2}$

Answer 1

$(x^2+14x+24)(x^2+11x+24)=4x^2 \ x^2(x+14+frac{24}{x})(x+11+frac{24}{x})=4x^2 |:x^2 \ (x+frac{24}{x}+14)(x+frac{24}{x}+11)=4 \ x+frac{24}{x} = t \ (t+14)(t+11)=4 \ t^2+11t+14t+154-4=0 \ t^2+25t+150=0 \ D=25^2-600 = 25 \ t_{1,2} = frac{-25pm5}{2} \ t_1 = frac{-25-5}{2} = -15 \ t_2 = frac{-25+5}{2} = -10 \ x+frac{24}{x} = -10 | * x \ x^2+10x+24=0 \ D = 100-24*4 = 4 \ x_{1,2} = frac{-10pm2}{2} \ x_1 = -6; x_2 = -4 \ x+frac{24}{x} = -15 \ x^2+15x+24=0 \ D = 225-24*4 = 129$
$x_{3,4} = frac{-15pmsqrt{129}}{2}$

$Ans: x in { -6, -4, frac{-15-sqrt{129}}{2}, frac{-15+sqrt{129}}{2}}$