Solve the equation by completing the square.
-$$9 m ^ { 2 } + 4 m = 0$$
A) 0
B) $\pm \frac { 4 } { 9 }$
C) $- \frac { 4 } { 9 } , 0$
D) $\frac { 4 } { 9 } , 0$

Question

Quizplus · Accepted Answer

We begin by moving all the terms to the left-hand side:

$$9m^2 + 4m = 0 \implies 9m^2 + 4m - 0 = 0$$

Next, we factor out any common coefficient from the terms involving the variable $m$. In this case, the common coefficient is $9$:

$$9(m^2 + \frac{4}{9} m) - 0 = 0$$

Now we focus on completing the square for the terms in the parentheses. We take half of the coefficient of $m$, square it, and add and subtract it inside the parentheses:

$$9(m^2 + \frac{4}{9} m + (\frac{2}{3})^2 - (\frac{2}{3})^2) - 0 = 0 $$

Simplifying this expression inside the parentheses gives:

$$ 9(m + \frac{2}{3})^2 - \frac{4}{3^2} \cdot 9 = 0 $$

Finally, we can solve for $m$:

$$9(m + \frac{2}{3})^2 = \frac{4}{9} \implies (m + \frac{2}{3})^2 = \frac{4}{9} \cdot \frac{1}{9} = \frac{4}{81}$$

Taking the square root of both sides and solving for $m$ gives:

$$m+\frac{2}{3} = \pm \frac{2}{9}$$

$$m = -\frac{2}{3} \pm \frac{2}{9} = - \frac{4}{9}, -\frac{2}{27}$$

Thus, our answer is $\boxed{	extbf{(C)} - \frac{4}{9}, 0}$.

Solve the Equation by Completing the Square 9m2+4m=09 m ^ { 2 } + 4 m = 09m2+4m=0

Solve the Equation by Completing the Square $9 m ^ { 2 } + 4 m = 0$