Solve for m 4m^2-6m=-360

$4 m^{2} - 6 m = - 360$

Step 1

Add $360$ to both sides of the equation.

$4 m^{2} - 6 m + 360 = 0$

Step 2

Factor $2$ out of $4 m^{2} - 6 m + 360$ .

Factor $2$ out of $4 m^{2}$ .

$2 (2 m^{2}) - 6 m + 360 = 0$

Factor $2$ out of $- 6 m$ .

$2 (2 m^{2}) + 2 (- 3 m) + 360 = 0$

Factor $2$ out of $360$ .

$2 (2 m^{2}) + 2 (- 3 m) + 2 (180) = 0$

Factor $2$ out of $2 (2 m^{2}) + 2 (- 3 m)$ .

$2 (2 m^{2} - 3 m) + 2 (180) = 0$

Factor $2$ out of $2 (2 m^{2} - 3 m) + 2 (180)$ .

$2 (2 m^{2} - 3 m + 180) = 0$

Step 3

Divide each term in $2 (2 m^{2} - 3 m + 180) = 0$ by $2$ and simplify.

Divide each term in $2 (2 m^{2} - 3 m + 180) = 0$ by $2$ .

$\frac{2 (2 m^{2} - 3 m + 180)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (2 m^{2} - 3 m + 180)}{2} = \frac{0}{2}$

Divide $2 m^{2} - 3 m + 180$ by $1$ .

$2 m^{2} - 3 m + 180 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$2 m^{2} - 3 m + 180 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 2$ , $b = - 3$ , and $c = 180$ into the quadratic formula and solve for $m$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot 180)}}{2 \cdot 2}$

Step 6

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$m = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot 180}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 180$ .

Multiply $- 4$ by $2$ .

$m = \frac{3 \pm \sqrt{9 - 8 \cdot 180}}{2 \cdot 2}$

Multiply $- 8$ by $180$ .

$m = \frac{3 \pm \sqrt{9 - 1440}}{2 \cdot 2}$

Subtract $1440$ from $9$ .

$m = \frac{3 \pm \sqrt{- 1431}}{2 \cdot 2}$

Rewrite $- 1431$ as $- 1 (1431)$ .

$m = \frac{3 \pm \sqrt{- 1 \cdot 1431}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (1431)}$ as $\sqrt{- 1} \cdot \sqrt{1431}$ .

$m = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{1431}}{2 \cdot 2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$m = \frac{3 \pm i \cdot \sqrt{1431}}{2 \cdot 2}$

Rewrite $1431$ as $3^{2} \cdot 159$ .

Factor $9$ out of $1431$ .

$m = \frac{3 \pm i \cdot \sqrt{9 (159)}}{2 \cdot 2}$

Rewrite $9$ as $3^{2}$ .

$m = \frac{3 \pm i \cdot \sqrt{3^{2} \cdot 159}}{2 \cdot 2}$

Pull terms out from under the radical.

$m = \frac{3 \pm i \cdot (3 \sqrt{159})}{2 \cdot 2}$

Move $3$ to the left of $i$ .

$m = \frac{3 \pm 3 i \sqrt{159}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$m = \frac{3 \pm 3 i \sqrt{159}}{4}$

Step 7

The final answer is the combination of both solutions.

$m = \frac{3 + 3 i \sqrt{159}}{4}, \frac{3 - 3 i \sqrt{159}}{4}$

Solve for m 4m^2-6m=-360

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