Algebra Examples

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

Step 1

Multiply through by the least common denominator $2$ , then simplify.

Tap for more steps...

Step 1.1

Apply the distributive property.

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Multiply $7$ by $2$ .

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

Step 1.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

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.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

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

Tap for more steps...

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 $2^{2} \cdot 51$ .

Tap for more steps...

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 $\pm$ .

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

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

Tap for more steps...

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 $2^{2} \cdot 51$ .

Tap for more steps...

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 $\pm$ .

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

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

Tap for more steps...

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 $2^{2} \cdot 51$ .

Tap for more steps...

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:

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

Enter YOUR Problem