Chapter 7.1, Problem 20OE

To determine

To Solve:

The equation by completing the square and by factoring:

  $t^2-6t=0$

Answer to Problem 20OE

  $t=\left\{0,6\right\}$

Explanation of Solution

Given:

The equation is :

  $t^2-6t=0$

Calculation:

Given:

  $t^2-6t=\mathrm{0.............}(1)$

First , we solve it by completing the square.

First , make the coefficient of $t^2$ 1.

Since here , it already is 1.

Check for the coefficient of t .

It is $-6$ .

So, the term that has to be added to make it a trinomial and a perfect square is :

square of the half of the coefficient of t

  $={\left(\frac{-6}{2}\right)}^2=9$

So, add 9 each side of the equation (1)

  $\begin{array}{l}{t}^2-6t+9=0+9\\ \Rightarrow t^2-6t+9=9\\ \Rightarrow {\left(t-3\right)}^2=\mathrm{9..............}[{\left(a-b\right)}^2=a^2-2ab+b^2]\\ \Rightarrow \sqrt{{\left(t-3\right)}^2}=\pm \sqrt{9}\\ \Rightarrow t-3=\pm 3\\ \Rightarrow t-3=3\text{ and }t-3=-3\\ \Rightarrow t=3+3\text{ and }t=-3+\mathrm{3...........}[\text{add 3 each side]}\\ \Rightarrow t=6\text{ and }t=0\\ \Rightarrow t=\left\{0,6\right\}\end{array}$