Algebra Examples

$5 - 10 m = 3 m^{2}$

Step 1

Subtract $3 m^{2}$ from both sides of the equation.

$5 - 10 m - 3 m^{2} = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 3$ , $b = - 10$ , and $c = 5$ into the quadratic formula and solve for $m$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 3 \cdot 5)}}{2 \cdot - 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 10$ to the power of $2$ .

$m = \frac{10 \pm \sqrt{100 - 4 \cdot - 3 \cdot 5}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4 \cdot - 3 \cdot 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 3$ .

$m = \frac{10 \pm \sqrt{100 + 12 \cdot 5}}{2 \cdot - 3}$

Step 4.1.2.2

Multiply $12$ by $5$ .

$m = \frac{10 \pm \sqrt{100 + 60}}{2 \cdot - 3}$

Step 4.1.3

Add $100$ and $60$ .

$m = \frac{10 \pm \sqrt{160}}{2 \cdot - 3}$

Step 4.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

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Step 4.1.4.1

Factor $16$ out of $160$ .

$m = \frac{10 \pm \sqrt{16 (10)}}{2 \cdot - 3}$

Step 4.1.4.2

Rewrite $16$ as $4^{2}$ .

$m = \frac{10 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot - 3}$

Step 4.1.5

Pull terms out from under the radical.

$m = \frac{10 \pm 4 \sqrt{10}}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$m = \frac{10 \pm 4 \sqrt{10}}{- 6}$

Step 4.3

Simplify $\frac{10 \pm 4 \sqrt{10}}{- 6}$ .

$m = \frac{5 \pm 2 \sqrt{10}}{- 3}$

Step 4.4

Move the negative in front of the fraction.

$m = - \frac{5 \pm 2 \sqrt{10}}{3}$

Step 5

The final answer is the combination of both solutions.

$m = - \frac{5 + 2 \sqrt{10}}{3}, - \frac{5 - 2 \sqrt{10}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$m = - \frac{5 + 2 \sqrt{10}}{3}, - \frac{5 - 2 \sqrt{10}}{3}$

Decimal Form:

$m = - 3.77485177 \dots, 0.44151844 \dots$