Algebra Examples

$m x^{2} + 12 x + 9 m = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = m$ , $b = 12$ , and $c = 9 m$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (m \cdot (9 m))}}{2 m}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Rewrite $4 \cdot m \cdot (9 m)$ as ${(6 m)}^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2} - {(6 m)}^{2}}}{2 m}$

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 12$ and $b = 6 m$ .

$x = \frac{- 12 \pm \sqrt{(12 + 6 m) (12 - (6 m))}}{2 m}$

Step 3.1.3

Simplify.

Tap for more steps...

Step 3.1.3.1

Factor $6$ out of $12 + 6 m$ .

Tap for more steps...

Step 3.1.3.1.1

Factor $6$ out of $12$ .

$x = \frac{- 12 \pm \sqrt{(6 \cdot 2 + 6 m) (12 - (6 m))}}{2 m}$

Step 3.1.3.1.2

Factor $6$ out of $6 \cdot 2 + 6 m$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (12 - (6 m))}}{2 m}$

Step 3.1.3.2

Multiply $6$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (12 - 6 m)}}{2 m}$

Step 3.1.4

Factor $6$ out of $12 - 6 m$ .

Tap for more steps...

Step 3.1.4.1

Factor $6$ out of $12$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2) - 6 m)}}{2 m}$

Step 3.1.4.2

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

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2) + 6 (- m))}}{2 m}$

Step 3.1.4.3

Factor $6$ out of $6 (2) + 6 (- m)$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2 - m))}}{2 m}$

$x = \frac{- 12 \pm \sqrt{6 (2 + m) \cdot (6 (2 - m))}}{2 m}$

Step 3.1.5

Multiply $6$ by $6$ .

$x = \frac{- 12 \pm \sqrt{36 (2 + m) (2 - m)}}{2 m}$

Step 3.1.6

Rewrite $36 (2 + m) (2 - m)$ as $6^{2} ((2 + m) (2 - m))$ .

Tap for more steps...

Step 3.1.6.1

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 12 \pm \sqrt{6^{2} (2 + m) (2 - m)}}{2 m}$

Step 3.1.6.2

Add parentheses.

$x = \frac{- 12 \pm \sqrt{6^{2} ((2 + m) (2 - m))}}{2 m}$

Step 3.1.7

Pull terms out from under the radical.

$x = \frac{- 12 \pm 6 \sqrt{(2 + m) (2 - m)}}{2 m}$

Step 3.2

Simplify $\frac{- 12 \pm 6 \sqrt{(2 + m) (2 - m)}}{2 m}$ .

$x = \frac{- 6 \pm 3 \sqrt{(2 + m) (2 - m)}}{m}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Rewrite $4 \cdot m \cdot (9 m)$ as ${(6 m)}^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2} - {(6 m)}^{2}}}{2 m}$

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 12$ and $b = 6 m$ .

$x = \frac{- 12 \pm \sqrt{(12 + 6 m) (12 - (6 m))}}{2 m}$

Step 4.1.3

Simplify.

Tap for more steps...

Step 4.1.3.1

Factor $6$ out of $12 + 6 m$ .

Tap for more steps...

Step 4.1.3.1.1

Factor $6$ out of $12$ .

$x = \frac{- 12 \pm \sqrt{(6 \cdot 2 + 6 m) (12 - (6 m))}}{2 m}$

Step 4.1.3.1.2

Factor $6$ out of $6 \cdot 2 + 6 m$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (12 - (6 m))}}{2 m}$

Step 4.1.3.2

Multiply $6$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (12 - 6 m)}}{2 m}$

Step 4.1.4

Factor $6$ out of $12 - 6 m$ .

Tap for more steps...

Step 4.1.4.1

Factor $6$ out of $12$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2) - 6 m)}}{2 m}$

Step 4.1.4.2

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

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2) + 6 (- m))}}{2 m}$

Step 4.1.4.3

Factor $6$ out of $6 (2) + 6 (- m)$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2 - m))}}{2 m}$

$x = \frac{- 12 \pm \sqrt{6 (2 + m) \cdot (6 (2 - m))}}{2 m}$

Step 4.1.5

Multiply $6$ by $6$ .

$x = \frac{- 12 \pm \sqrt{36 (2 + m) (2 - m)}}{2 m}$

Step 4.1.6

Rewrite $36 (2 + m) (2 - m)$ as $6^{2} ((2 + m) (2 - m))$ .

Tap for more steps...

Step 4.1.6.1

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 12 \pm \sqrt{6^{2} (2 + m) (2 - m)}}{2 m}$

Step 4.1.6.2

Add parentheses.

$x = \frac{- 12 \pm \sqrt{6^{2} ((2 + m) (2 - m))}}{2 m}$

Step 4.1.7

Pull terms out from under the radical.

$x = \frac{- 12 \pm 6 \sqrt{(2 + m) (2 - m)}}{2 m}$

Step 4.2

Simplify $\frac{- 12 \pm 6 \sqrt{(2 + m) (2 - m)}}{2 m}$ .

$x = \frac{- 6 \pm 3 \sqrt{(2 + m) (2 - m)}}{m}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{- 6 + 3 \sqrt{(2 + m) (2 - m)}}{m}$

Step 4.4

Factor $3$ out of $- 6 + 3 \sqrt{(2 + m) (2 - m)}$ .

Tap for more steps...

Step 4.4.1

Factor $3$ out of $- 6$ .

$x = \frac{3 \cdot - 2 + 3 \sqrt{(2 + m) (2 - m)}}{m}$

Step 4.4.2

Factor $3$ out of $3 \cdot - 2 + 3 \sqrt{(2 + m) (2 - m)}$ .

$x = \frac{3 (- 2 + \sqrt{(2 + m) (2 - m)})}{m}$

Step 4.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{3 (- 1 \cdot 2 + \sqrt{(2 + m) (2 - m)})}{m}$

Step 4.6

Factor $- 1$ out of $\sqrt{(2 + m) (2 - m)}$ .

$x = \frac{3 (- 1 \cdot 2 - 1 (- \sqrt{(2 + m) (2 - m)}))}{m}$

Step 4.7

Factor $- 1$ out of $- 1 (2) - 1 (- \sqrt{(2 + m) (2 - m)})$ .

$x = \frac{3 (- 1 (2 - \sqrt{(2 + m) (2 - m)}))}{m}$

Step 4.8

Move the negative in front of the fraction.

$x = - \frac{3 (2 - \sqrt{(2 + m) (2 - m)})}{m}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Rewrite $4 \cdot m \cdot (9 m)$ as ${(6 m)}^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2} - {(6 m)}^{2}}}{2 m}$

Step 5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 12$ and $b = 6 m$ .

$x = \frac{- 12 \pm \sqrt{(12 + 6 m) (12 - (6 m))}}{2 m}$

Step 5.1.3

Simplify.

Tap for more steps...

Step 5.1.3.1

Factor $6$ out of $12 + 6 m$ .

Tap for more steps...

Step 5.1.3.1.1

Factor $6$ out of $12$ .

$x = \frac{- 12 \pm \sqrt{(6 \cdot 2 + 6 m) (12 - (6 m))}}{2 m}$

Step 5.1.3.1.2

Factor $6$ out of $6 \cdot 2 + 6 m$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (12 - (6 m))}}{2 m}$

Step 5.1.3.2

Multiply $6$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (12 - 6 m)}}{2 m}$

Step 5.1.4

Factor $6$ out of $12 - 6 m$ .

Tap for more steps...

Step 5.1.4.1

Factor $6$ out of $12$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2) - 6 m)}}{2 m}$

Step 5.1.4.2

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

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2) + 6 (- m))}}{2 m}$

Step 5.1.4.3

Factor $6$ out of $6 (2) + 6 (- m)$ .

$x = \frac{- 12 \pm \sqrt{6 (2 + m) (6 (2 - m))}}{2 m}$

$x = \frac{- 12 \pm \sqrt{6 (2 + m) \cdot (6 (2 - m))}}{2 m}$

Step 5.1.5

Multiply $6$ by $6$ .

$x = \frac{- 12 \pm \sqrt{36 (2 + m) (2 - m)}}{2 m}$

Step 5.1.6

Rewrite $36 (2 + m) (2 - m)$ as $6^{2} ((2 + m) (2 - m))$ .

Tap for more steps...

Step 5.1.6.1

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 12 \pm \sqrt{6^{2} (2 + m) (2 - m)}}{2 m}$

Step 5.1.6.2

Add parentheses.

$x = \frac{- 12 \pm \sqrt{6^{2} ((2 + m) (2 - m))}}{2 m}$

Step 5.1.7

Pull terms out from under the radical.

$x = \frac{- 12 \pm 6 \sqrt{(2 + m) (2 - m)}}{2 m}$

Step 5.2

Simplify $\frac{- 12 \pm 6 \sqrt{(2 + m) (2 - m)}}{2 m}$ .

$x = \frac{- 6 \pm 3 \sqrt{(2 + m) (2 - m)}}{m}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{- 6 - 3 \sqrt{(2 + m) (2 - m)}}{m}$

Step 5.4

Factor $- 3$ out of $- 6 - 3 \sqrt{(2 + m) (2 - m)}$ .

Tap for more steps...

Step 5.4.1

Reorder the expression.

Tap for more steps...

Step 5.4.1.1

Reorder $2$ and $m$ .

$x = \frac{- 6 - 3 \sqrt{(m + 2) (2 - m)}}{m}$

Step 5.4.1.2

Reorder $2$ and $- m$ .

$x = \frac{- 6 - 3 \sqrt{(m + 2) (- m + 2)}}{m}$

Step 5.4.1.3

Reorder $- 6$ and $- 3 \sqrt{(m + 2) (- m + 2)}$ .

$x = \frac{- 3 \sqrt{(m + 2) (- m + 2)} - 6}{m}$

Step 5.4.2

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

$x = \frac{- 3 \sqrt{(m + 2) (- m + 2)} - 6}{m}$

Step 5.4.3

Factor $- 3$ out of $- 6$ .

$x = \frac{- 3 \sqrt{(m + 2) (- m + 2)} - 3 \cdot 2}{m}$

Step 5.4.4

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

$x = \frac{- 3 (\sqrt{(m + 2) (- m + 2)} + 2)}{m}$

Step 5.5

Move the negative in front of the fraction.

$x = - \frac{3 (\sqrt{(m + 2) (- m + 2)} + 2)}{m}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{3 (2 - \sqrt{(2 + m) (2 - m)})}{m}$

$x = - \frac{3 (\sqrt{(m + 2) (- m + 2)} + 2)}{m}$