Algebra Examples

x^{2} + 7 x - \frac{1}{2} = 0

$x^{2} + 7 x - \frac{1}{2} = 0$

Step 1

Multiply through by the least common denominator

2

$2$ , then simplify.

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Step 1.1

Apply the distributive property.

2 x^{2} + 2 (7 x) + 2 (- \frac{1}{2}) = 0

$2 x^{2} + 2 (7 x) + 2 (- \frac{1}{2}) = 0$

Step 1.2

Simplify.

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Step 1.2.1

Multiply

7

$7$ by

2

$2$ .

2 x^{2} + 14 x + 2 (- \frac{1}{2}) = 0

$2 x^{2} + 14 x + 2 (- \frac{1}{2}) = 0$

Step 1.2.2

Cancel the common factor of

2

$2$ .

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Step 1.2.2.1

Move the leading negative in

- \frac{1}{2}

$- \frac{1}{2}$ into the numerator.

2 x^{2} + 14 x + 2 (\frac{- 1}{2}) = 0

$2 x^{2} + 14 x + 2 (\frac{- 1}{2}) = 0$

Step 1.2.2.2

Cancel the common factor.

$2 x^{2} + 14 x + 2 (\frac{- 1}{2}) = 0$

Step 1.2.2.3

Rewrite the expression.

$2 x^{2} + 14 x - 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values

$a = 2$ ,

$b = 14$ , and

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

$x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

$14$ to the power of

$2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 4.1.2

Multiply

$- 4 \cdot 2 \cdot - 1$ .

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Step 4.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 1}}{2 \cdot 2}$

Step 4.1.2.2

Multiply

$- 8$ by

$- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 8}}{2 \cdot 2}$

Step 4.1.3

Add

$196$ and

$8$ .

$x = \frac{- 14 \pm \sqrt{204}}{2 \cdot 2}$

Step 4.1.4

Rewrite

$204$ as

$2^{2} \cdot 51$ .

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Step 4.1.4.1

Factor

$4$ out of

$204$ .

$x = \frac{- 14 \pm \sqrt{4 (51)}}{2 \cdot 2}$

Step 4.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 51}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{51}}{2 \cdot 2}$

Step 4.2

Multiply

$2$ by

$2$ .

$x = \frac{- 14 \pm 2 \sqrt{51}}{4}$

Step 4.3

Simplify

$\frac{- 14 \pm 2 \sqrt{51}}{4}$ .

$x = \frac{- 7 \pm \sqrt{51}}{2}$

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

$14$ to the power of

$2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 5.1.2

Multiply

$- 4 \cdot 2 \cdot - 1$ .

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Step 5.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 1}}{2 \cdot 2}$

Step 5.1.2.2

Multiply

$- 8$ by

$- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 8}}{2 \cdot 2}$

Step 5.1.3

Add

$196$ and

$8$ .

$x = \frac{- 14 \pm \sqrt{204}}{2 \cdot 2}$

Step 5.1.4

Rewrite

$204$ as

$2^{2} \cdot 51$ .

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Step 5.1.4.1

Factor

$4$ out of

$204$ .

$x = \frac{- 14 \pm \sqrt{4 (51)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 51}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{51}}{2 \cdot 2}$

Step 5.2

Multiply

$2$ by

$2$ .

$x = \frac{- 14 \pm 2 \sqrt{51}}{4}$

Step 5.3

Simplify

$\frac{- 14 \pm 2 \sqrt{51}}{4}$ .

$x = \frac{- 7 \pm \sqrt{51}}{2}$

Step 5.4

Change the

$\pm$ to

$+$ .

$x = \frac{- 7 + \sqrt{51}}{2}$

Step 5.5

Rewrite

$- 7$ as

$- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{51}}{2}$

Step 5.6

Factor

$- 1$ out of

$\sqrt{51}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{51})}{2}$

Step 5.7

Factor

$- 1$ out of

$- 1 (7) - 1 (- \sqrt{51})$ .

$x = \frac{- 1 (7 - \sqrt{51})}{2}$

Step 5.8

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{51}}{2}$

Step 6

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise

$14$ to the power of

$2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 6.1.2

Multiply

$- 4 \cdot 2 \cdot - 1$ .

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Step 6.1.2.1

Multiply

$- 4$ by

$2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 1}}{2 \cdot 2}$

Step 6.1.2.2

Multiply

$- 8$ by

$- 1$ .

$x = \frac{- 14 \pm \sqrt{196 + 8}}{2 \cdot 2}$

Step 6.1.3

Add

$196$ and

$8$ .

$x = \frac{- 14 \pm \sqrt{204}}{2 \cdot 2}$

Step 6.1.4

Rewrite

$204$ as

$2^{2} \cdot 51$ .

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Step 6.1.4.1

Factor

$4$ out of

$204$ .

$x = \frac{- 14 \pm \sqrt{4 (51)}}{2 \cdot 2}$

Step 6.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 51}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{51}}{2 \cdot 2}$

Step 6.2

Multiply

$2$ by

$2$ .

$x = \frac{- 14 \pm 2 \sqrt{51}}{4}$

Step 6.3

Simplify

$\frac{- 14 \pm 2 \sqrt{51}}{4}$ .

$x = \frac{- 7 \pm \sqrt{51}}{2}$

Step 6.4

Change the

$\pm$ to

$-$ .

$x = \frac{- 7 - \sqrt{51}}{2}$

Step 6.5

Rewrite

$- 7$ as

$- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{51}}{2}$

Step 6.6

Factor

$- 1$ out of

$- \sqrt{51}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{51})}{2}$

Step 6.7

Factor

$- 1$ out of

$- 1 (7) - (\sqrt{51})$ .

$x = \frac{- 1 (7 + \sqrt{51})}{2}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{51}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{51}}{2}, - \frac{7 + \sqrt{51}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{51}}{2}, - \frac{7 + \sqrt{51}}{2}$

Decimal Form:

$x = 0.07071421 \dots, - 7.07071421 \dots$

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