Question

Solve the equation

$\Rightarrow 16={\left(\frac{1+5\lambda }{1+\lambda }\right)}^2+{\left(\frac{2+6\lambda }{1+\lambda }\right)}^2+\frac{40\lambda -4}{1+\lambda }$

Answer 1

$16={\left(\frac{1+5\lambda }{1+\lambda }\right)}^2+{\left(\frac{2+6\lambda }{1+\lambda }\right)}^2+\frac{40\lambda -4}{1+\lambda }$

$16(1+\lambda {)}^2=(1+5\lambda {)}^2+(2+6\lambda {)}^2+(40\lambda -4\left)\right(1+\lambda )$

$16+16{\lambda }^2+32\lambda =1+25{\lambda }^2+10\lambda +4+36{\lambda }^2+24\lambda +40{\lambda }^2+40\lambda -4-4\lambda$

$16{\lambda }^2+32\lambda +16=101{\lambda }^2+70\lambda +1$

$85{\lambda }^2+38\lambda -15=0$

On solving

$\lambda =\frac{-38\pm \sqrt{(38{)}^2+60\times 85}}{170}$

$\lambda =0.252,-0.699$