Algebra Examples

$x^{2} - 8 x + 11 = 0$

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 $a = 1$ , $b = - 8$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 11}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{8 \pm \sqrt{64 - 44}}{2 \cdot 1}$

Step 3.1.3

Subtract $44$ from $64$ .

$x = \frac{8 \pm \sqrt{20}}{2 \cdot 1}$

Step 3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.1.4.1

Factor $4$ out of $20$ .

$x = \frac{8 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 3.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{5}}{2}$

Step 3.3

Simplify $\frac{8 \pm 2 \sqrt{5}}{2}$ .

$x = 4 \pm \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 4 \pm \sqrt{5}$

Decimal Form:

$x = 6.23606797 \dots, 1.76393202 \dots$