Precalculus Examples

$v = x (12 - 2 x)$

Step 1

Rewrite the equation as $x (12 - 2 x) = v$ .

$x (12 - 2 x) = v$

Step 2

Simplify $x (12 - 2 x)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$x \cdot 12 + x (- 2 x) = v$

Step 2.1.2

Reorder.

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Step 2.1.2.1

Move $12$ to the left of $x$ .

$12 \cdot x + x (- 2 x) = v$

Step 2.1.2.2

Rewrite using the commutative property of multiplication.

$12 \cdot x - 2 x \cdot x = v$

Step 2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1

Move $x$ .

$12 x - 2 (x \cdot x) = v$

Step 2.2.2

Multiply $x$ by $x$ .

$12 x - 2 x^{2} = v$

Step 3

Subtract $v$ from both sides of the equation.

$12 x - 2 x^{2} - v = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 2$ , $b = 12$ , and $c = - v$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 2 \cdot (- 1 v)}}{2 \cdot - 2}$

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 12 \pm \sqrt{144 + 8 \cdot (- 1 v)}}{2 \cdot - 2}$

Step 6.1.2.2

Multiply $8$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 - 8 v}}{2 \cdot - 2}$

Step 6.1.3

Factor $8$ out of $144 - 8 v$ .

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Step 6.1.3.1

Factor $8$ out of $144$ .

$x = \frac{- 12 \pm \sqrt{8 (18) - 8 v}}{2 \cdot - 2}$

Step 6.1.3.2

Factor $8$ out of $- 8 v$ .

$x = \frac{- 12 \pm \sqrt{8 (18) + 8 (- v)}}{2 \cdot - 2}$

Step 6.1.3.3

Factor $8$ out of $8 (18) + 8 (- v)$ .

$x = \frac{- 12 \pm \sqrt{8 (18 - v)}}{2 \cdot - 2}$

Step 6.1.4

Rewrite $8 (18 - v)$ as $2^{2} \cdot (2 (18 - v))$ .

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Step 6.1.4.1

Factor $4$ out of $8$ .

$x = \frac{- 12 \pm \sqrt{4 (2) (18 - v)}}{2 \cdot - 2}$

Step 6.1.4.2

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - v))}}{2 \cdot - 2}$

Step 6.1.4.3

Add parentheses.

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - v))}}{2 \cdot - 2}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{2 \cdot - 2}$

Step 6.2

Multiply $2$ by $- 2$ .

$x = \frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{- 4}$

Step 6.3

Simplify $\frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{- 4}$ .

$x = \frac{6 \pm \sqrt{2 (18 - v)}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 2 \cdot (- 1 v)}}{2 \cdot - 2}$

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 12 \pm \sqrt{144 + 8 \cdot (- 1 v)}}{2 \cdot - 2}$

Step 7.1.2.2

Multiply $8$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 - 8 v}}{2 \cdot - 2}$

Step 7.1.3

Factor $8$ out of $144 - 8 v$ .

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Step 7.1.3.1

Factor $8$ out of $144$ .

$x = \frac{- 12 \pm \sqrt{8 (18) - 8 v}}{2 \cdot - 2}$

Step 7.1.3.2

Factor $8$ out of $- 8 v$ .

$x = \frac{- 12 \pm \sqrt{8 (18) + 8 (- v)}}{2 \cdot - 2}$

Step 7.1.3.3

Factor $8$ out of $8 (18) + 8 (- v)$ .

$x = \frac{- 12 \pm \sqrt{8 (18 - v)}}{2 \cdot - 2}$

Step 7.1.4

Rewrite $8 (18 - v)$ as $2^{2} \cdot (2 (18 - v))$ .

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Step 7.1.4.1

Factor $4$ out of $8$ .

$x = \frac{- 12 \pm \sqrt{4 (2) (18 - v)}}{2 \cdot - 2}$

Step 7.1.4.2

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - v))}}{2 \cdot - 2}$

Step 7.1.4.3

Add parentheses.

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - v))}}{2 \cdot - 2}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{2 \cdot - 2}$

Step 7.2

Multiply $2$ by $- 2$ .

$x = \frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{- 4}$

Step 7.3

Simplify $\frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{- 4}$ .

$x = \frac{6 \pm \sqrt{2 (18 - v)}}{2}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{6 + \sqrt{2 (18 - v)}}{2}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 2 \cdot (- 1 v)}}{2 \cdot - 2}$

Step 8.1.2

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

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Step 8.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 12 \pm \sqrt{144 + 8 \cdot (- 1 v)}}{2 \cdot - 2}$

Step 8.1.2.2

Multiply $8$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 - 8 v}}{2 \cdot - 2}$

Step 8.1.3

Factor $8$ out of $144 - 8 v$ .

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Step 8.1.3.1

Factor $8$ out of $144$ .

$x = \frac{- 12 \pm \sqrt{8 (18) - 8 v}}{2 \cdot - 2}$

Step 8.1.3.2

Factor $8$ out of $- 8 v$ .

$x = \frac{- 12 \pm \sqrt{8 (18) + 8 (- v)}}{2 \cdot - 2}$

Step 8.1.3.3

Factor $8$ out of $8 (18) + 8 (- v)$ .

$x = \frac{- 12 \pm \sqrt{8 (18 - v)}}{2 \cdot - 2}$

Step 8.1.4

Rewrite $8 (18 - v)$ as $2^{2} \cdot (2 (18 - v))$ .

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Step 8.1.4.1

Factor $4$ out of $8$ .

$x = \frac{- 12 \pm \sqrt{4 (2) (18 - v)}}{2 \cdot - 2}$

Step 8.1.4.2

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - v))}}{2 \cdot - 2}$

Step 8.1.4.3

Add parentheses.

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - v))}}{2 \cdot - 2}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{2 \cdot - 2}$

Step 8.2

Multiply $2$ by $- 2$ .

$x = \frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{- 4}$

Step 8.3

Simplify $\frac{- 12 \pm 2 \sqrt{2 (18 - v)}}{- 4}$ .

$x = \frac{6 \pm \sqrt{2 (18 - v)}}{2}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{6 - \sqrt{2 (18 - v)}}{2}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{6 + \sqrt{2 (18 - v)}}{2}$

$x = \frac{6 - \sqrt{2 (18 - v)}}{2}$