Algebra Examples

x^{2} - 4 x - 1 = 0

$x^{2} - 4 x - 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values

a = 1

$a = 1$ ,

b = - 4

$b = - 4$ , and

c = - 1

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

x

$x$ .

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

- 4

$- 4$ to the power of

2

$2$ .

x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.1.2

Multiply

- 4 \cdot 1 \cdot - 1

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

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1}}{2 \cdot 1}

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.1.2.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

x = \frac{4 \pm \sqrt{16 + 4}}{2 \cdot 1}

$x = \frac{4 \pm \sqrt{16 + 4}}{2 \cdot 1}$

x = \frac{4 \pm \sqrt{16 + 4}}{2 \cdot 1}

$x = \frac{4 \pm \sqrt{16 + 4}}{2 \cdot 1}$

Step 3.1.3

Add

16

$16$ and

4

$4$ .

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

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

Step 3.1.4

Rewrite

20

$20$ as

2^{2} \cdot 5

$2^{2} \cdot 5$ .

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Step 3.1.4.1

Factor

4

$4$ out of

20

$20$ .

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

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

Step 3.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 3.1.5

Pull terms out from under the radical.

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

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

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

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

Step 3.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{4 \pm 2 \sqrt{5}}{2}

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

Step 3.3

Simplify

\frac{4 \pm 2 \sqrt{5}}{2}

$\frac{4 \pm 2 \sqrt{5}}{2}$ .

x = 2 \pm \sqrt{5}

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

x = 2 \pm \sqrt{5}

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

Step 4

The result can be shown in multiple forms.

Exact Form:

x = 2 \pm \sqrt{5}

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

Decimal Form:

x = 4.23606797 \dots, - 0.23606797 \dots

$x = 4.23606797 \dots, - 0.23606797 \dots$