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

Subtract

4

$4$ from

16

$16$ .

x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 3.1.4

Rewrite

12

$12$ as

2^{2} \cdot 3

$2^{2} \cdot 3$ .

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Step 3.1.4.1

Factor

4

$4$ out of

12

$12$ .

x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 3.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 3.1.5

Pull terms out from under the radical.

x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 3.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{- 4 \pm 2 \sqrt{3}}{2}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2}$

Step 3.3

Simplify

\frac{- 4 \pm 2 \sqrt{3}}{2}

$\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

x = - 2 \pm \sqrt{3}

$x = - 2 \pm \sqrt{3}$

x = - 2 \pm \sqrt{3}

$x = - 2 \pm \sqrt{3}$

Step 4

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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 4.1.2

Multiply

- 4 \cdot 1 \cdot 1

$- 4 \cdot 1 \cdot 1$ .

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Step 4.1.2.1

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 4.1.2.2

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 4.1.3

Subtract

4

$4$ from

16

$16$ .

x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 4.1.4

Rewrite

12

$12$ as

2^{2} \cdot 3

$2^{2} \cdot 3$ .

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Step 4.1.4.1

Factor

4

$4$ out of

12

$12$ .

x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 4.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 4.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{- 4 \pm 2 \sqrt{3}}{2}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2}$

Step 4.3

Simplify

\frac{- 4 \pm 2 \sqrt{3}}{2}

$\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

x = - 2 \pm \sqrt{3}

$x = - 2 \pm \sqrt{3}$

Step 4.4

Change the

\pm

$\pm$ to

+

$+$ .

x = - 2 + \sqrt{3}

$x = - 2 + \sqrt{3}$

x = - 2 + \sqrt{3}

$x = - 2 + \sqrt{3}$

Step 5

Simplify the expression to solve for the

-

$-$ portion of the

\pm

$\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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 5.1.2

Multiply

- 4 \cdot 1 \cdot 1

$- 4 \cdot 1 \cdot 1$ .

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Step 5.1.2.1

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 5.1.2.2

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 5.1.3

Subtract

4

$4$ from

16

$16$ .

x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{12}}{2 \cdot 1}$

Step 5.1.4

Rewrite

12

$12$ as

2^{2} \cdot 3

$2^{2} \cdot 3$ .

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Step 5.1.4.1

Factor

4

$4$ out of

12

$12$ .

x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 5.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{- 4 \pm 2 \sqrt{3}}{2}

$x = \frac{- 4 \pm 2 \sqrt{3}}{2}$

Step 5.3

Simplify

\frac{- 4 \pm 2 \sqrt{3}}{2}

$\frac{- 4 \pm 2 \sqrt{3}}{2}$ .

x = - 2 \pm \sqrt{3}

$x = - 2 \pm \sqrt{3}$

Step 5.4

Change the

\pm

$\pm$ to

-

$-$ .

x = - 2 - \sqrt{3}

$x = - 2 - \sqrt{3}$

x = - 2 - \sqrt{3}

$x = - 2 - \sqrt{3}$

Step 6

The final answer is the combination of both solutions.

x = - 2 + \sqrt{3}, - 2 - \sqrt{3}

$x = - 2 + \sqrt{3}, - 2 - \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

x = - 2 + \sqrt{3}, - 2 - \sqrt{3}

$x = - 2 + \sqrt{3}, - 2 - \sqrt{3}$

Decimal Form:

x = - 0.26794919 \dots, - 3.73205080 \dots

$x = - 0.26794919 \dots, - 3.73205080 \dots$