Chapter 12.2, Problem 13WE

a.

To determine

To Solve: The equation $x^2-12x+35=0$ by completing the square.

Answer to Problem 13WE

The required solution set is $\left\{7,5\right\}$ .

Explanation of Solution

Given information:

  $x^2-12x+35=0$

Calculation:

  $\begin{array}{l}{x}^2-12x+35=0\\ x^2-12x=-35\\ x^2-12x+{\left(-\frac{12}{2}\right)}^2=-35+{\left(-\frac{12}{2}\right)}^2\end{array}$

(Add square of half the coefficient of x to both sides)

  $\begin{array}{l}{x}^2-12x+36=-35+36\\ {\left(x-6\right)}^2=1\end{array}$

(Simplify)

  $\begin{array}{l}x-6=\sqrt{1}\\ x-6=\pm 1\end{array}$

(Take square root on both sides)

  $x=6\pm 1$

(Add 6 to each side)

  $\begin{array}{l}x=6+1\\ x=7\end{array}$

Or

  $\begin{array}{l}x=6-1\\ x=5\end{array}$

Therefore, the solution set is $\left\{7,5\right\}$ .

b.

To determine

To Solve: The equation $x^2-12x+35=0$ by factoring.

Answer to Problem 13WE

The required solution set is $\left\{7,5\right\}$ .

Explanation of Solution

Given information:

  $x^2-12x+35=0$

Calculation:

  $\begin{array}{l}{x}^2-12x+35=0\\ x^2-7x-5x+35=0\\ x\left(x-7\right)-5\left(x-7\right)=0\\ \left(x-7\right)\left(x-5\right)=0\end{array}$

(Splitting the middle term and factoring)

Therefore,

  $\begin{array}{l}x=7\\ or\\ x=5\end{array}$

So, the solution set is $\left\{7,5\right\}$ .