Algebra Examples

$2 m (m - 5) + 10 = 0$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$2 m \cdot m + 2 m \cdot - 5 + 10 = 0$

Step 1.2

Multiply $m$ by $m$ by adding the exponents.

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Step 1.2.1

Move $m$ .

$2 (m \cdot m) + 2 m \cdot - 5 + 10 = 0$

Step 1.2.2

Multiply $m$ by $m$ .

$2 m^{2} + 2 m \cdot - 5 + 10 = 0$

Step 1.3

Multiply $- 5$ by $2$ .

$2 m^{2} - 10 m + 10 = 0$

Step 2

Factor $2$ out of $2 m^{2} - 10 m + 10$ .

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Step 2.1

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

$2 (m^{2}) - 10 m + 10 = 0$

Step 2.2

Factor $2$ out of $- 10 m$ .

$2 (m^{2}) + 2 (- 5 m) + 10 = 0$

Step 2.3

Factor $2$ out of $10$ .

$2 m^{2} + 2 (- 5 m) + 2 \cdot 5 = 0$

Step 2.4

Factor $2$ out of $2 m^{2} + 2 (- 5 m)$ .

$2 (m^{2} - 5 m) + 2 \cdot 5 = 0$

Step 2.5

Factor $2$ out of $2 (m^{2} - 5 m) + 2 \cdot 5$ .

$2 (m^{2} - 5 m + 5) = 0$

Step 3

Divide each term in $2 (m^{2} - 5 m + 5) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (m^{2} - 5 m + 5) = 0$ by $2$ .

$\frac{2 (m^{2} - 5 m + 5)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (m^{2} - 5 m + 5)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $m^{2} - 5 m + 5$ by $1$ .

$m^{2} - 5 m + 5 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$m^{2} - 5 m + 5 = 0$

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 $a = 1$ , $b = - 5$ , and $c = 5$ into the quadratic formula and solve for $m$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 5$ to the power of $2$ .

$m = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$m = \frac{5 \pm \sqrt{25 - 4 \cdot 5}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $5$ .

$m = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 1}$

Step 6.1.3

Subtract $20$ from $25$ .

$m = \frac{5 \pm \sqrt{5}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$m = \frac{5 \pm \sqrt{5}}{2}$

Step 7

The final answer is the combination of both solutions.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$m = 3.61803398 \dots, 1.38196601 \dots$