The number of real roots of the equation $\sqrt{1+\sqrt{5}x+5x^2} + \sqrt{1-\sqrt{5}x+5x^2}=4$, is _____.
Answer & explanation
We have,
$\sqrt{1+\sqrt{5}x+5x^2} + \sqrt{1-\sqrt{5}x+5x^2}=4$ ...(i)
Also,
$(1+\sqrt{5}x+5x^2)-(1-\sqrt{5}x+5x^2) = 2 \sqrt{5} x$ ...(ii)
Dividing (ii) by (i), we get
$\sqrt{1+\sqrt{5}x+5x^2}-\sqrt{1-\sqrt{5}x+5x^2}=\frac{\sqrt{5}}{2}x$ ...(iii)
Adding (i) and (iii), we get
$\sqrt{1+\sqrt{5}x+5x^2}-2+\frac{\sqrt{5}}{4}x$ ...(iii)
$⇒1+\sqrt{5}x+5x^2=4+\frac{5}{16}x^2+\sqrt{5} x$
$⇒x^2=\frac{16}{25}⇒x±\frac{4}{5}$