Question

The sum of a number and its positive square root is $\frac{6}{25}$. Find the number.

Answer 1

$$\begin{gathered} \begin{array}{l}\text{Let the required number be}x.\\ \text{According to the question:}\end{array} \\ x\text{+}\sqrt{x}=\frac{6}{25} \\ \Rightarrow 25x+25\sqrt{x}=6 \\ \Rightarrow 25x+25\sqrt{x}-6=0 \\ \text{Let}\sqrt{x}\text{be equal to}y. \\ \therefore 25y^2+25y-6=0 \\ \Rightarrow 25y^2+(30-5)y-6=0 \\ \Rightarrow 25y^2+30y-5y-6=0 \\ \Rightarrow 5y(5y+6)-1(5y+6)=0 \\ \Rightarrow (5y+6)(5y-1)=0 \\ \Rightarrow \text{5}y\text{+ 6 = 0 or 5}y-1=0 \\ \Rightarrow y=\frac{-6}{5}\text{or}y=\frac{1}{5} \\ \Rightarrow y=\frac{1}{5}(\because y\text{cannot be negative}) \\ \Rightarrow \sqrt{x}=\frac{1}{5} \\ \Rightarrow x=\frac{1}{25} \\ \text{Hence, the required number is}\frac{1}{25}\text{.} \end{gathered}$$