Question

The sum of a number and its squares is $\frac{63}{4}$. Find the number.

Answer 1

$$\begin{gathered} \text{Let the required number be}x. \\ \text{According to the question:} \\ x\text{+}{x}^2=\frac{63}{4} \\ \Rightarrow 4x+4x^2=63 \\ \Rightarrow 4x^2+4x-63=0 \\ \Rightarrow 4x^2+(18-14)x-63=0 \\ \Rightarrow 4x^2+18x-14x-63=0 \\ \Rightarrow 2x(2x+9)-7(2x+9)=0 \\ \Rightarrow (2x+9)(2x-7)=0 \\ \Rightarrow \text{2}x+9=0\text{or 2}x-7=0 \\ \Rightarrow x=\frac{-9}{2}\text{or}x=\frac{7}{2} \\ \text{Hence, the number is either}\frac{-9}{2}\text{or}\frac{7}{2}. \end{gathered}$$