Practicing Success
The number of real roots of the equation $\sqrt{1+\sqrt{5}x+5x^2} + \sqrt{1-\sqrt{5}x+5x^2}=4$, is _____. |
2 |
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) $⇒1+\sqrt{5}x+5x^2=4+\frac{5}{16}x^2+\sqrt{5} x$ $⇒x^2=\frac{16}{25}⇒x±\frac{4}{5}$ |