Question

If the distance between the points (2, –2) and (–1, x) is 5, then the values of x are __________.

Answer 1

The distance between $\left(x_1,y_1\right)\mathrm{and}\left(x_2,y_2\right)\mathrm{is}\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Therefore, the distance between the points (2, –2) and (–1, x)$=\sqrt{{\left(-1-2\right)}^2+{\left(x-\left(-2\right)\right)}^2}$
$$\begin{gathered} \Rightarrow 5=\sqrt{{\left(-3\right)}^2+{\left(x+2\right)}^2} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow {\left(5\right)}^2={\left(\sqrt{9+{\left(x+2\right)}^2}\right)}^2 \\ \Rightarrow 25=9+{\left(x+2\right)}^2 \\ \Rightarrow 25=9+x^2+4+4x \\ \Rightarrow x^2+4x+13-25=0 \\ \Rightarrow x^2+4x-12=0 \\ \Rightarrow x^2+6x-2x-12=0 \\ \Rightarrow x\left(x+6\right)-2\left(x+6\right)=0 \\ \Rightarrow \left(x-2\right)\left(x+6\right)=0 \\ \Rightarrow x=2,-6 \end{gathered}$$

Hence, the values of x are 2 and ​–6.