Chapter SH, Problem 4.12EP

To determine

Find the solution of the given equation using the ‘completing the square’ method.

Answer to Problem 4.12EP

The solutions of the given equation are $x=-4\text{and}x=6$ .

Explanation of Solution

Given:

The given equationis $x^2-2x-24=0$ .

Calculation:

  $x^2-2x-24=0$

Add $24$ to both sides.

  $\begin{array}{l}\Rightarrow x^2-2x-24+24=0+24\\ \Rightarrow x^2-2x=24\end{array}$

Add $1$ to both sides.

  $\begin{array}{l}\Rightarrow x^2-2x+1=24+1\\ \Rightarrow x^2-2x+1=25\end{array}$

Apply identity $x^2\pm 2xy+y^2={\left(x\pm y\right)}^2$ in the left side.

  $\begin{array}{l}\Rightarrow x^2+2\cdot 1\cdot x+1^2=25\\ \Rightarrow {\left(x-1\right)}^2=25\\ \Rightarrow x-1=\pm \sqrt{25}\\ \Rightarrow x-1=\pm 5\\ \Rightarrow x-1=5\text{and}\Rightarrow x-1=-5\\ \Rightarrow x-1+1=5+1\text{and}\Rightarrow x-1+1=-5+1\\ \Rightarrow x=6\text{and}\Rightarrow x=-4\end{array}$

Hence thesolutions of the given equation are $x=-4\text{and}x=6$ .