Algebra Examples

$3 q - 8 = \frac{6}{q}$

Step 1

Subtract $\frac{6}{q}$ from both sides of the equation.

$3 q - 8 - \frac{6}{q} = 0$

Step 2

Reorder terms.

$3 q - \frac{6}{q} - 8 = 0$

Step 3

To write $3 q$ as a fraction with a common denominator, multiply by $\frac{q}{q}$ .

$\frac{3 q \cdot q}{q} - \frac{6}{q} - 8 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{3 q \cdot q - 6}{q} - 8 = 0$

Step 5

Simplify the numerator.

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Step 5.1

Factor $3$ out of $3 q \cdot q - 6$ .

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Step 5.1.1

Factor $3$ out of $3 q \cdot q$ .

$\frac{3 (q \cdot q) - 6}{q} - 8 = 0$

Step 5.1.2

Factor $3$ out of $- 6$ .

$\frac{3 (q \cdot q) + 3 \cdot - 2}{q} - 8 = 0$

Step 5.1.3

Factor $3$ out of $3 (q \cdot q) + 3 \cdot - 2$ .

$\frac{3 (q \cdot q - 2)}{q} - 8 = 0$

Step 5.2

Multiply $q$ by $q$ .

$\frac{3 (q^{2} - 2)}{q} - 8 = 0$

Step 6

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{q}{q}$ .

$\frac{3 (q^{2} - 2)}{q} - 8 \cdot \frac{q}{q} = 0$

Step 7

Combine $- 8$ and $\frac{q}{q}$ .

$\frac{3 (q^{2} - 2)}{q} + \frac{- 8 q}{q} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{3 (q^{2} - 2) - 8 q}{q} = 0$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{3 q^{2} + 3 \cdot - 2 - 8 q}{q} = 0$

Step 9.2

Multiply $3$ by $- 2$ .

$\frac{3 q^{2} - 6 - 8 q}{q} = 0$

Step 9.3

Reorder terms.

$\frac{3 q^{2} - 8 q - 6}{q} = 0$

Step 10

Set the numerator equal to zero.

$3 q^{2} - 8 q - 6 = 0$

Step 11

Solve the equation for $q$ .

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Step 11.1

Use the quadratic formula to find the solutions.

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

Step 11.2

Substitute the values $a = 3$ , $b = - 8$ , and $c = - 6$ into the quadratic formula and solve for $q$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot - 6)}}{2 \cdot 3}$

Step 11.3

Simplify.

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Step 11.3.1

Simplify the numerator.

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Step 11.3.1.1

Raise $- 8$ to the power of $2$ .

$q = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 6}}{2 \cdot 3}$

Step 11.3.1.2

Multiply $- 4 \cdot 3 \cdot - 6$ .

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Step 11.3.1.2.1

Multiply $- 4$ by $3$ .

$q = \frac{8 \pm \sqrt{64 - 12 \cdot - 6}}{2 \cdot 3}$

Step 11.3.1.2.2

Multiply $- 12$ by $- 6$ .

$q = \frac{8 \pm \sqrt{64 + 72}}{2 \cdot 3}$

Step 11.3.1.3

Add $64$ and $72$ .

$q = \frac{8 \pm \sqrt{136}}{2 \cdot 3}$

Step 11.3.1.4

Rewrite $136$ as $2^{2} \cdot 34$ .

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Step 11.3.1.4.1

Factor $4$ out of $136$ .

$q = \frac{8 \pm \sqrt{4 (34)}}{2 \cdot 3}$

Step 11.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$q = \frac{8 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 3}$

Step 11.3.1.5

Pull terms out from under the radical.

$q = \frac{8 \pm 2 \sqrt{34}}{2 \cdot 3}$

Step 11.3.2

Multiply $2$ by $3$ .

$q = \frac{8 \pm 2 \sqrt{34}}{6}$

Step 11.3.3

Simplify $\frac{8 \pm 2 \sqrt{34}}{6}$ .

$q = \frac{4 \pm \sqrt{34}}{3}$

Step 11.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 11.4.1

Simplify the numerator.

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Step 11.4.1.1

Raise $- 8$ to the power of $2$ .

$q = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 6}}{2 \cdot 3}$

Step 11.4.1.2

Multiply $- 4 \cdot 3 \cdot - 6$ .

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Step 11.4.1.2.1

Multiply $- 4$ by $3$ .

$q = \frac{8 \pm \sqrt{64 - 12 \cdot - 6}}{2 \cdot 3}$

Step 11.4.1.2.2

Multiply $- 12$ by $- 6$ .

$q = \frac{8 \pm \sqrt{64 + 72}}{2 \cdot 3}$

Step 11.4.1.3

Add $64$ and $72$ .

$q = \frac{8 \pm \sqrt{136}}{2 \cdot 3}$

Step 11.4.1.4

Rewrite $136$ as $2^{2} \cdot 34$ .

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Step 11.4.1.4.1

Factor $4$ out of $136$ .

$q = \frac{8 \pm \sqrt{4 (34)}}{2 \cdot 3}$

Step 11.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$q = \frac{8 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 3}$

Step 11.4.1.5

Pull terms out from under the radical.

$q = \frac{8 \pm 2 \sqrt{34}}{2 \cdot 3}$

Step 11.4.2

Multiply $2$ by $3$ .

$q = \frac{8 \pm 2 \sqrt{34}}{6}$

Step 11.4.3

Simplify $\frac{8 \pm 2 \sqrt{34}}{6}$ .

$q = \frac{4 \pm \sqrt{34}}{3}$

Step 11.4.4

Change the $\pm$ to $+$ .

$q = \frac{4 + \sqrt{34}}{3}$

Step 11.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.5.1

Simplify the numerator.

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Step 11.5.1.1

Raise $- 8$ to the power of $2$ .

$q = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 6}}{2 \cdot 3}$

Step 11.5.1.2

Multiply $- 4 \cdot 3 \cdot - 6$ .

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Step 11.5.1.2.1

Multiply $- 4$ by $3$ .

$q = \frac{8 \pm \sqrt{64 - 12 \cdot - 6}}{2 \cdot 3}$

Step 11.5.1.2.2

Multiply $- 12$ by $- 6$ .

$q = \frac{8 \pm \sqrt{64 + 72}}{2 \cdot 3}$

Step 11.5.1.3

Add $64$ and $72$ .

$q = \frac{8 \pm \sqrt{136}}{2 \cdot 3}$

Step 11.5.1.4

Rewrite $136$ as $2^{2} \cdot 34$ .

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Step 11.5.1.4.1

Factor $4$ out of $136$ .

$q = \frac{8 \pm \sqrt{4 (34)}}{2 \cdot 3}$

Step 11.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$q = \frac{8 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot 3}$

Step 11.5.1.5

Pull terms out from under the radical.

$q = \frac{8 \pm 2 \sqrt{34}}{2 \cdot 3}$

Step 11.5.2

Multiply $2$ by $3$ .

$q = \frac{8 \pm 2 \sqrt{34}}{6}$

Step 11.5.3

Simplify $\frac{8 \pm 2 \sqrt{34}}{6}$ .

$q = \frac{4 \pm \sqrt{34}}{3}$

Step 11.5.4

Change the $\pm$ to $-$ .

$q = \frac{4 - \sqrt{34}}{3}$

Step 11.6

The final answer is the combination of both solutions.

$q = \frac{4 + \sqrt{34}}{3}, \frac{4 - \sqrt{34}}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$q = \frac{4 + \sqrt{34}}{3}, \frac{4 - \sqrt{34}}{3}$

Decimal Form:

$q = 3.27698396 \dots, - 0.61031729 \dots$