Solve for q 3q-8=6/q

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

Add $8$ to both sides of the equation.

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

Find the LCD of the terms in the equation.

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

$1, q, 1$

The LCM of one and any expression is the expression.

$q$

Multiply each term in $3 q = \frac{6}{q} + 8$ by $q$ to eliminate the fractions.

Multiply each term in $3 q = \frac{6}{q} + 8$ by $q$ .

$3 q \cdot q = \frac{6}{q} q + 8 q$

Simplify the left side.

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

Move $q$ .

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

Multiply $q$ by $q$ .

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

Simplify the right side.

Cancel the common factor of $q$ .

Cancel the common factor.

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

Rewrite the expression.

$3 q^{2} = 6 + 8 q$

Solve the equation.

Subtract $8 q$ from both sides of the equation.

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

Subtract $6$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $3$ .

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

Multiply $- 12$ by $- 6$ .

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

Add $64$ and $72$ .

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

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

Factor $4$ out of $136$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $3$ .

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

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

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

The final answer is the combination of both solutions.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

Solve for q 3q-8=6/q

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