9.6.2 Choosing the Best Method to Solve an Equation

The correct answers are: 2.5, -0.5.

Compare your answer:

Completing the square method:

$\begin{array}{ccc}\hfill x^2-2x-1.25& \hfill =\hfill & 0\hfill \\ \hfill x^2-2x& \hfill =\hfill & 1.25\hfill \\ \hfill x^2-2x+{\left(\frac{-2}{2}\right)}^2& \hfill =\hfill & 1.25+{\left(\frac{-2}{2}\right)}^2\hfill \\ \hfill x^2-2x+1& \hfill =\hfill & 1.25+1\hfill \\ \hfill x^2-2x+1& \hfill =\hfill & 2.25\hfill \\ \hfill (x-1{)}^2& \hfill =\hfill & 2.25\hfill \\ \hfill \sqrt{(x-1{)}^2}& \hfill =\hfill & \sqrt{2.25}\hfill \\ \hfill x-1& \hfill =\hfill & \pm \sqrt{2.25}\hfill \\ \hfill x& \hfill =\hfill & 1\pm \sqrt{2.25}\hfill \\ \hfill x& \hfill =\hfill & 1\pm 1.5\hfill \\ \hfill x& \hfill =\hfill & 1+1.5\hfill \\ \hfill x& \hfill =\hfill & 1-1.5\hfill \\ \hfill x& \hfill =\hfill & 2.5\hfill \\ \hfill x& \hfill =\hfill & -0.5\hfill \\ \hfill x& \hfill =\hfill & 2.5\text{ and }x=-0.5\hfill \end{array}$

The correct answers are: 2.2, -4.

Compare your answer:

Completing the square method:

$\begin{array}{ccc}\hfill 5x^2+9x-44& \hfill =\hfill & 0\hfill \\ \hfill 5x^2+9x& \hfill =\hfill & 44\hfill \\ \hfill \frac{5x^2+9x}{5}& \hfill =\hfill & \frac{44}{5}\hfill \\ \hfill x^2+\frac{9}{5}x+{\left(\frac{1}{2}·\frac{9}{5}\right)}^2& \hfill =\hfill & \frac{44}{5}+{\left(\frac{1}{2}·\frac{9}{5}\right)}^2\hfill \\ \hfill x^2+\frac{9}{5}x+\frac{81}{100}& \hfill =\hfill & \frac{44}{5}+\frac{81}{100}\hfill \\ \hfill x^2+\frac{9}{5}x+\frac{81}{100}& \hfill =\hfill & \frac{961}{100}\hfill \\ \hfill {(x+\frac{9}{10})}^2& \hfill =\hfill & \frac{961}{100}\hfill \\ \hfill \sqrt{{(x+\frac{9}{10})}^2}& \hfill =\hfill & \sqrt{\frac{961}{100}}\hfill \\ \hfill x+\frac{9}{10}& \hfill =\hfill & \pm \frac{31}{10}\hfill \\ \hfill x& \hfill =\hfill & -\frac{9}{10}\pm \frac{31}{10}\hfill \\ \hfill x& \hfill =\hfill & -\frac{9}{10}+\frac{31}{10}\hfill \\ \hfill x& \hfill =\hfill & -\frac{9}{10}-\frac{31}{10}\hfill \\ \hfill x& \hfill =\hfill & \frac{22}{10}=2.2\hfill \\ \hfill x& \hfill =\hfill & -\frac{40}{10}=-4\hfill \\ \hfill x& \hfill =\hfill & 2.2\text{ and }x=-4\hfill \end{array}$

The correct answers are: 0.25, -1.5.

Compare your answer:

Completing the square method:

$$\begin{gathered} x^2+1.25x=0.375 \\ {x}^2+1.25x+((0.5)(1.25){)}^2=0.375+((0.5)(1.25)){}^2 \\ {x}^2+1.25x+0.390625=0.375+0.390625 \\ (x+0.625{)}^2=0.765625 \\ \sqrt{(x+0.625{)}^2}=\sqrt{0.765625} \\ x+0.625=\pm 0.875 \\ x=-0.625\pm 0.875 \\ x=-0.625+0.875 \\ x=-0.625-0.875 \\ x=0.25 \\ x=-1.5 \\ x=0.25\text{ and }x=-1.5 \end{gathered}$$

The correct answers are: $\frac{7}{2}+\sqrt{5}$, $\frac{7}{2}-\sqrt{5}$.

Compare your answer:

Completing the square method:

$\begin{array}{ccc}\hfill 4x^2-28x+29& \hfill =\hfill & 0\hfill \\ \hfill \frac{4x^2-28x+29}{4}& \hfill =\hfill & 0\hfill \\ \hfill x^2-7x+\frac{29}{4}& \hfill =\hfill & 0\hfill \\ \hfill x^2-7x& \hfill =\hfill & -\frac{29}{4}\hfill \\ \hfill x^2-7x+{\left(\frac{7}{2}\right)}^2& \hfill =\hfill & -\frac{29}{4}+{\left(\frac{7}{2}\right)}^2\hfill \\ \hfill x^2-7x+\frac{49}{4}& \hfill =\hfill & \frac{20}{4}\hfill \\ \hfill x^2-7x+\frac{49}{4}& \hfill =\hfill & 5\hfill \\ \hfill {(x-\frac{7}{2})}^2& \hfill =\hfill & 5\hfill \\ \hfill \sqrt{{(x-\frac{7}{2})}^2}& \hfill =\hfill & \sqrt{5}\hfill \\ \hfill x-\frac{7}{2}& \hfill =\hfill & \pm \sqrt{5}\hfill \\ \hfill x& \hfill =\hfill & \frac{7}{2}\pm \sqrt{5}\hfill \\ \hfill x& \hfill =\hfill & \frac{7}{2}+\sqrt{5}\hfill \\ \hfill x& \hfill =\hfill & \frac{7}{2}-\sqrt{5}\hfill \\ \hfill x& \hfill =\hfill & \frac{7}{2}+\sqrt{5}\text{ and }x=\frac{7}{2}-\sqrt{5}\hfill \end{array}$