Algebra Examples

x^{2} - 2 x - 1 = 0

$x^{2} - 2 x - 1 = 0$

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

$c = - 1$ into the quadratic formula and solve for

$x$ .

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

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

$2$ .

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

Step 3.1.2

Multiply

$- 4 \cdot 1 \cdot - 1$ .

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Step 3.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 3.1.2.2

Multiply

$- 4$ by

$- 1$ .

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

Step 3.1.3

Add

$4$ and

$4$ .

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

Step 3.1.4

Rewrite

$8$ as

$2^{2} \cdot 2$ .

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Step 3.1.4.1

Factor

$4$ out of

$8$ .

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

Step 3.1.4.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 3.1.5

Pull terms out from under the radical.

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

Step 3.2

Multiply

$2$ by

$1$ .

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

Step 3.3

Simplify

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

$x = 1 \pm \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 1 \pm \sqrt{2}$

Decimal Form:

$x = 2.41421356 \dots, - 0.41421356 \dots$