Chapter 12.2, Problem 15WE

a.

To determine

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

Answer to Problem 15WE

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

Explanation of Solution

Given information:

  $z^2-2z-35=0$

Calculation:

  $\begin{array}{l}{z}^2-2z-35=0\\ z^2-2z=35\end{array}$

(Add 35 to each side and rewrite the equation)

  $z^2-2z+{\left(-1\right)}^2=35+{\left(-1\right)}^2$

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

  $\begin{array}{l}{z}^2-2z+1=36\\ {\left(z-1\right)}^2=36\end{array}$

(Simplify)

  $\begin{array}{l}z-1=\sqrt{36}\\ z-1=\pm 6\end{array}$

(Perform square root on both sides)

  $z=1\pm 6$

(Add 1 to each side)

So,

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

Or

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

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

b.

To determine

To Solve: The equation $z^2-2z-35=0$ by factoring.

Answer to Problem 15WE

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

Explanation of Solution

Given information:

  $z^2-2z-35=0$

Calculation:

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

(Splitting the middle term and factoring)

Therefore,

  $\begin{array}{l}z=7\\ orz=-5\end{array}$

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