Question

The sum of four times a number and three times its reciprocal is 7. Find that number.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{number}\mathrm{be}x. \\ \text{Four}\mathrm{times}\text{the number}\mathrm{is}4x\mathrm{and}\mathrm{its}\mathrm{reciprocal}\mathrm{is}\frac{1}{x}. \\ \mathrm{Thus},\mathrm{by}\mathrm{the}\mathrm{given}\mathrm{condition},\mathrm{we}\mathrm{get}: \\ 4x+\frac{3}{x}=7 \\ \Rightarrow \frac{4x^2+3}{x}=7 \end{gathered}$$

$\Rightarrow$ 4x²+3 = 7x
⇒ 4x²$-$7x+3 = 0
On splitting the middle term $-$7x as $-$4x$-$3x, we get:
⇒ 4x²$-$4x$-$3x+3 = 0

$$\begin{gathered} \Rightarrow 4x\left(x-1\right)-3\left(x-1\right)=0 \\ \Rightarrow \left(x-1\right)\left(4x-3\right)=0 \\ \Rightarrow x-1=0\mathrm{or}4x-3=0 \\ \Rightarrow x=1\mathrm{or}x=\frac{3}{4} \\ \mathrm{Thus}\mathrm{the}\mathrm{number}\mathrm{is}\mathrm{either}1\mathrm{or}\frac{3}{4} \end{gathered}$$