Algebra Examples

$\frac{- (- 10) + - \sqrt{{(- 10)}^{2} - 4 \cdot 3 \cdot - 1}}{2 (3)}$

Step 1

Replace all occurrences of $+$ $-$ with a single $-$ . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because $1 \cdot - 1 = - 1$

$\frac{- (- 10) - \sqrt{{(- 10)}^{2} - 4 \cdot 3 \cdot - 1}}{2 (3)}$

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $- 10$ .

$\frac{10 - \sqrt{{(- 10)}^{2} - 4 \cdot 3 \cdot - 1}}{2 (3)}$

Step 2.2

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

$\frac{10 - \sqrt{100 - 4 \cdot 3 \cdot - 1}}{2 (3)}$

Step 2.3

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 2.3.1

Multiply $- 4$ by $3$ .

$\frac{10 - \sqrt{100 - 12 \cdot - 1}}{2 (3)}$

Step 2.3.2

Multiply $- 12$ by $- 1$ .

$\frac{10 - \sqrt{100 + 12}}{2 (3)}$

Step 2.4

Add $100$ and $12$ .

$\frac{10 - \sqrt{112}}{2 (3)}$

Step 2.5

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 2.5.1

Factor $16$ out of $112$ .

$\frac{10 - \sqrt{16 (7)}}{2 (3)}$

Step 2.5.2

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

$\frac{10 - \sqrt{4^{2} \cdot 7}}{2 (3)}$

Step 2.6

Pull terms out from under the radical.

$\frac{10 - (4 \sqrt{7})}{2 (3)}$

Step 2.7

Multiply $4$ by $- 1$ .

$\frac{10 - 4 \sqrt{7}}{2 (3)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Multiply $2$ by $3$ .

$\frac{10 - 4 \sqrt{7}}{6}$

Step 3.2

Cancel the common factor of $10 - 4 \sqrt{7}$ and $6$ .

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Step 3.2.1

Factor $2$ out of $10$ .

$\frac{2 (5) - 4 \sqrt{7}}{6}$

Step 3.2.2

Factor $2$ out of $- 4 \sqrt{7}$ .

$\frac{2 (5) + 2 (- 2 \sqrt{7})}{6}$

Step 3.2.3

Factor $2$ out of $2 (5) + 2 (- 2 \sqrt{7})$ .

$\frac{2 (5 - 2 \sqrt{7})}{6}$

Step 3.2.4

Cancel the common factors.

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Step 3.2.4.1

Factor $2$ out of $6$ .

$\frac{2 (5 - 2 \sqrt{7})}{2 \cdot 3}$

Step 3.2.4.2

Cancel the common factor.

$\frac{2 (5 - 2 \sqrt{7})}{2 \cdot 3}$

Step 3.2.4.3

Rewrite the expression.

$\frac{5 - 2 \sqrt{7}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{5 - 2 \sqrt{7}}{3}$

Decimal Form:

$- 0.09716754 \dots$