Choosing the Best Method to Solve an Equation

9.6.2 Choosing the Best Method to Solve an Equation

9.6.2 • Choosing the Best Method to Solve an Equation

Activity

Write down your answer for how to solve each equation. Find the two solutions by rewriting it in factored form or by completing the square. Be prepared to explain your choice of method.

Select two solutions for $x^{2} - 2 x - 1.25 = 0$ .

2.3
3
–1
–0.5

The correct answers are: 2.5, -0.5.

Compare your answer:

Completing the square method:

$\begin{matrix} x^{2} - 2 x - 1.25 & = & 0 \\ x^{2} - 2 x & = & 1.25 \\ x^{2} - 2 x + {(\frac{- 2}{2})}^{2} & = & 1.25 + {(\frac{- 2}{2})}^{2} \\ x^{2} - 2 x + 1 & = & 1.25 + 1 \\ x^{2} - 2 x + 1 & = & 2.25 \\ (x - 1)^{2} & = & 2.25 \\ \sqrt{(x - 1)^{2}} & = & \sqrt{2.25} \\ x - 1 & = & \pm \sqrt{2.25} \\ x & = & 1 \pm \sqrt{2.25} \\ x & = & 1 \pm 1.5 \\ x & = & 1 + 1.5 \\ x & = & 1 - 1.5 \\ x & = & 2.5 \\ x & = & - 0.5 \\ x & = & 2.5 and x = - 0.5 \end{matrix}$

Select two solutions for $5 x^{2} + 9 x - 44 = 0$ .

2.2
–20
–4
11

The correct answers are: 2.2, -4.

Compare your answer:

Completing the square method:

$\begin{matrix} 5 x^{2} + 9 x - 44 & = & 0 \\ 5 x^{2} + 9 x & = & 44 \\ \frac{5 x^{2} + 9 x}{5} & = & \frac{44}{5} \\ x^{2} + \frac{9}{5} x + {(\frac{1}{2} \cdot \frac{9}{5})}^{2} & = & \frac{44}{5} + {(\frac{1}{2} \cdot \frac{9}{5})}^{2} \\ x^{2} + \frac{9}{5} x + \frac{81}{100} & = & \frac{44}{5} + \frac{81}{100} \\ x^{2} + \frac{9}{5} x + \frac{81}{100} & = & \frac{961}{100} \\ {(x + \frac{9}{10})}^{2} & = & \frac{961}{100} \\ \sqrt{{(x + \frac{9}{10})}^{2}} & = & \sqrt{\frac{961}{100}} \\ x + \frac{9}{10} & = & \pm \frac{31}{10} \\ x & = & - \frac{9}{10} \pm \frac{31}{10} \\ x & = & - \frac{9}{10} + \frac{31}{10} \\ x & = & - \frac{9}{10} - \frac{31}{10} \\ x & = & \frac{22}{10} = 2.2 \\ x & = & - \frac{40}{10} = - 4 \\ x & = & 2.2 and x = - 4 \end{matrix}$

Select two solutions for $x^{2} + 1.25 x = 0.375$ .

0.75
0.25
–1.5
2.5

The correct answers are: 0.25, -1.5.

Compare your answer:

Completing the square method:

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

Select two solutions for $4 x^{2} - 28 x + 29 = 0$ .

$- 7 + \sqrt{2}$
$7 - \sqrt{2}$
$\frac{7}{2} + \sqrt{5}$
$\frac{7}{2} - \sqrt{5}$

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

Compare your answer:

Completing the square method:

$\begin{matrix} 4 x^{2} - 28 x + 29 & = & 0 \\ \frac{4 x^{2} - 28 x + 29}{4} & = & 0 \\ x^{2} - 7 x + \frac{29}{4} & = & 0 \\ x^{2} - 7 x & = & - \frac{29}{4} \\ x^{2} - 7 x + {(\frac{7}{2})}^{2} & = & - \frac{29}{4} + {(\frac{7}{2})}^{2} \\ x^{2} - 7 x + \frac{49}{4} & = & \frac{20}{4} \\ x^{2} - 7 x + \frac{49}{4} & = & 5 \\ {(x - \frac{7}{2})}^{2} & = & 5 \\ \sqrt{{(x - \frac{7}{2})}^{2}} & = & \sqrt{5} \\ x - \frac{7}{2} & = & \pm \sqrt{5} \\ x & = & \frac{7}{2} \pm \sqrt{5} \\ x & = & \frac{7}{2} + \sqrt{5} \\ x & = & \frac{7}{2} - \sqrt{5} \\ x & = & \frac{7}{2} + \sqrt{5} and x = \frac{7}{2} - \sqrt{5} \end{matrix}$

Self Check

Use completing the square to solve

2x^2+10x+8=0

$x=-5 \pm \sqrt{17}$
$x=-4, x=-1$
$x= 1, x =4$
$x= \frac{25}{4}$

Additional Resources

Solve by Using Perfect Squares and Completing the Square

Solve $2 x^{2} + 5 x + 2 = 0$ by completing the square.

Step 1 - Divide all terms by the leading coefficient.

$\frac{2 x^{2} + 5 x + 2}{2} = \frac{0}{2}$ $x^{2} + \frac{5}{2} x + 1 = 0$

Step 2 - Move the constant to the right side of the equation.

$x^{2} + \frac{5}{2} x = - 1$

Step 3 - Complete the square and add to each side: $(\frac{5}{2} \times \frac{1}{2})^{2} = \frac{25}{16}$ .

$x^{2} + \frac{5}{2} x + \frac{25}{16} = - 1 + \frac{25}{16}$

Step 4 - Simplify. (Hint: create a common denominator before adding.)

$x^{2} + \frac{5}{2} x + \frac{25}{16} = - \frac{16}{16} + \frac{25}{16}$

$(x + \frac{5}{4})^{2} = \frac{9}{16}$

Step 5 - Take the square root of each side.

$\sqrt{(x + \frac{5}{4})^{2}} = \sqrt{\frac{9}{16}}$

$x + \frac{5}{4} = \pm \frac{3}{4}$

Step 6 - Write two equations and solve.

$x + \frac{5}{4} = \frac{3}{4}$ $x + \frac{5}{4} = - \frac{3}{4}$

$x = - \frac{2}{4} = - \frac{1}{2}$ $x = - \frac{8}{4} = - 2$

Try it

Solve by Using Perfect Squares and Completing the Square

Solve $4 x^{2} + 36 x + 37 = 0$ .

Here is how to solve using completing the square:

Step 1 - Divide all terms by the leading coefficient.

$\frac{4 x^{2} + 36 x + 37}{4} = \frac{0}{4}$

$x^{2} + 9 x + \frac{37}{4} = 0$

Step 2 - Move the constant to the right side of the equation.

$x^{2} + 9 x = - \frac{37}{4}$

Step 3 - Complete the square and add to each side:

$(9 \times \frac{1}{2})^{2} = \frac{81}{4}$ .

$x^{2} + 9 x + \frac{81}{4} = - \frac{37}{4} + \frac{81}{4}$

Step 4 - Simplify.

$(x + \frac{9}{2})^{2} = 11$

Step 5 - Take the square root of each side.

$\sqrt{(x + \frac{9}{2})^{2}} = \sqrt{11}$

$(x + \frac{9}{2}) = \pm \sqrt{11}$

Step 6 - Write two equations and solve.

$x + \frac{9}{2} = \sqrt{11}$ $x + \frac{9}{2} = - \sqrt{11}$

$x = - \frac{9}{2} + \sqrt{11}$ $x = - \frac{9}{2} - \sqrt{11}$