Solve for q 12q-6=1/q

$12 q - 6 = \frac{1}{q}$

Add $6$ to both sides of the equation.

$12 q = \frac{1}{q} + 6$

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 $12 q = \frac{1}{q} + 6$ by $q$ to eliminate the fractions.

Multiply each term in $12 q = \frac{1}{q} + 6$ by $q$ .

$12 q \cdot q = \frac{1}{q} q + 6 q$

Simplify the left side.

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

Move $q$ .

$12 (q \cdot q) = \frac{1}{q} q + 6 q$

Multiply $q$ by $q$ .

$12 q^{2} = \frac{1}{q} q + 6 q$

Simplify the right side.

Cancel the common factor of $q$ .

Cancel the common factor.

$12 q^{2} = \frac{1}{q} q + 6 q$

Rewrite the expression.

$12 q^{2} = 1 + 6 q$

Solve the equation.

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

$12 q^{2} - 6 q = 1$

Subtract $1$ from both sides of the equation.

$12 q^{2} - 6 q - 1 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 12$ , $b = - 6$ , and $c = - 1$ into the quadratic formula and solve for $q$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (12 \cdot - 1)}}{2 \cdot 12}$

Simplify.

Simplify the numerator.

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

$q = \frac{6 \pm \sqrt{36 - 4 \cdot 12 \cdot - 1}}{2 \cdot 12}$

Multiply $- 4$ by $12$ .

$q = \frac{6 \pm \sqrt{36 - 48 \cdot - 1}}{2 \cdot 12}$

Multiply $- 48$ by $- 1$ .

$q = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 12}$

Add $36$ and $48$ .

$q = \frac{6 \pm \sqrt{84}}{2 \cdot 12}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$q = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 12}$

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

$q = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 12}$

Pull terms out from under the radical.

$q = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 12}$

Multiply $2$ by $12$ .

$q = \frac{6 \pm 2 \sqrt{21}}{24}$

Simplify $\frac{6 \pm 2 \sqrt{21}}{24}$ .

$q = \frac{3 \pm \sqrt{21}}{12}$

The final answer is the combination of both solutions.

$q = \frac{3 + \sqrt{21}}{12}, \frac{3 - \sqrt{21}}{12}$

The result can be shown in multiple forms.

Exact Form:

$q = \frac{3 + \sqrt{21}}{12}, \frac{3 - \sqrt{21}}{12}$

Decimal Form:

$q = 0.63188130 \dots, - 0.13188130 \dots$

Solve for q 12q-6=1/q

Related Questions

Support LCMGCF.com by disabling your adblocker.