Algebra Examples

$x^{2} - 2 x - 5 = 0$

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 $a = 1$ , $b = - 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 20}}{2 \cdot 1}$

Step 3.1.3

Add $4$ and $20$ .

$x = \frac{2 \pm \sqrt{24}}{2 \cdot 1}$

Step 3.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.1.4.1

Factor $4$ out of $24$ .

$x = \frac{2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 3.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{6}}{2}$

Step 3.3

Simplify $\frac{2 \pm 2 \sqrt{6}}{2}$ .

$x = 1 \pm \sqrt{6}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 20}}{2 \cdot 1}$

Step 4.1.3

Add $4$ and $20$ .

$x = \frac{2 \pm \sqrt{24}}{2 \cdot 1}$

Step 4.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.1.4.1

Factor $4$ out of $24$ .

$x = \frac{2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 4.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{6}}{2}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{6}}{2}$ .

$x = 1 \pm \sqrt{6}$

Step 4.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{6}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 20}}{2 \cdot 1}$

Step 5.1.3

Add $4$ and $20$ .

$x = \frac{2 \pm \sqrt{24}}{2 \cdot 1}$

Step 5.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 5.1.4.1

Factor $4$ out of $24$ .

$x = \frac{2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{6}}{2}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{6}}{2}$ .

$x = 1 \pm \sqrt{6}$

Step 5.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{6}$

Step 6

The final answer is the combination of both solutions.

$x = 1 + \sqrt{6}, 1 - \sqrt{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{6}, 1 - \sqrt{6}$

Decimal Form:

$x = 3.44948974 \dots, - 1.44948974 \dots$