Finite Math Examples

$11 q - 6 = \frac{2}{q}$

Step 1

Add $6$ to both sides of the equation.

$11 q = \frac{2}{q} + 6$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, q, 1$

Step 2.2

The LCM of one and any expression is the expression.

$q$

Step 3

Multiply each term in $11 q = \frac{2}{q} + 6$ by $q$ to eliminate the fractions.

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Step 3.1

Multiply each term in $11 q = \frac{2}{q} + 6$ by $q$ .

$11 q \cdot q = \frac{2}{q} q + 6 q$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $q$ by $q$ by adding the exponents.

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Step 3.2.1.1

Move $q$ .

$11 (q \cdot q) = \frac{2}{q} q + 6 q$

Step 3.2.1.2

Multiply $q$ by $q$ .

$11 q^{2} = \frac{2}{q} q + 6 q$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $q$ .

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Step 3.3.1.1

Cancel the common factor.

$11 q^{2} = \frac{2}{q} q + 6 q$

Step 3.3.1.2

Rewrite the expression.

$11 q^{2} = 2 + 6 q$

Step 4

Solve the equation.

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Step 4.1

Subtract $6 q$ from both sides of the equation.

$11 q^{2} - 6 q = 2$

Step 4.2

Subtract $2$ from both sides of the equation.

$11 q^{2} - 6 q - 2 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 11$ , $b = - 6$ , and $c = - 2$ into the quadratic formula and solve for $q$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (11 \cdot - 2)}}{2 \cdot 11}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $- 6$ to the power of $2$ .

$q = \frac{6 \pm \sqrt{36 - 4 \cdot 11 \cdot - 2}}{2 \cdot 11}$

Step 4.5.1.2

Multiply $- 4 \cdot 11 \cdot - 2$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $11$ .

$q = \frac{6 \pm \sqrt{36 - 44 \cdot - 2}}{2 \cdot 11}$

Step 4.5.1.2.2

Multiply $- 44$ by $- 2$ .

$q = \frac{6 \pm \sqrt{36 + 88}}{2 \cdot 11}$

Step 4.5.1.3

Add $36$ and $88$ .

$q = \frac{6 \pm \sqrt{124}}{2 \cdot 11}$

Step 4.5.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 4.5.1.4.1

Factor $4$ out of $124$ .

$q = \frac{6 \pm \sqrt{4 (31)}}{2 \cdot 11}$

Step 4.5.1.4.2

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

$q = \frac{6 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 11}$

Step 4.5.1.5

Pull terms out from under the radical.

$q = \frac{6 \pm 2 \sqrt{31}}{2 \cdot 11}$

Step 4.5.2

Multiply $2$ by $11$ .

$q = \frac{6 \pm 2 \sqrt{31}}{22}$

Step 4.5.3

Simplify $\frac{6 \pm 2 \sqrt{31}}{22}$ .

$q = \frac{3 \pm \sqrt{31}}{11}$

Step 4.6

The final answer is the combination of both solutions.

$q = \frac{3 + \sqrt{31}}{11}, \frac{3 - \sqrt{31}}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$q = \frac{3 + \sqrt{31}}{11}, \frac{3 - \sqrt{31}}{11}$

Decimal Form:

$q = 0.77888766 \dots, - 0.23343312 \dots$