Precalculus Examples

$v = 2 (a b + b c + c a)$

Step 1

Rewrite the equation as $2 (a b + b c + c a) = v$ .

$2 (a b + b c + c a) = v$

Step 2

Divide each term in $2 (a b + b c + c a) = v$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (a b + b c + c a) = v$ by $2$ .

$\frac{2 (a b + b c + c a)}{2} = \frac{v}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 (a b + b c + c a)}{2} = \frac{v}{2}$

Step 2.2.1.2

Divide $a b + b c + c a$ by $1$ .

$a b + b c + c a = \frac{v}{2}$

Step 3

Subtract $b c$ from both sides of the equation.

$a b + c a = \frac{v}{2} - b c$

Step 4

Factor $a$ out of $a b + c a$ .

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Step 4.1

Factor $a$ out of $a b$ .

$a (b) + c a = \frac{v}{2} - b c$

Step 4.2

Factor $a$ out of $c a$ .

$a (b) + a c = \frac{v}{2} - b c$

Step 4.3

Factor $a$ out of $a (b) + a c$ .

$a (b + c) = \frac{v}{2} - b c$

Step 5

Divide each term in $a (b + c) = \frac{v}{2} - b c$ by $b + c$ and simplify.

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Step 5.1

Divide each term in $a (b + c) = \frac{v}{2} - b c$ by $b + c$ .

$\frac{a (b + c)}{b + c} = \frac{\frac{v}{2}}{b + c} + \frac{- b c}{b + c}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $b + c$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{a (b + c)}{b + c} = \frac{\frac{v}{2}}{b + c} + \frac{- b c}{b + c}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{\frac{v}{2}}{b + c} + \frac{- b c}{b + c}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$a = \frac{\frac{v}{2} - b c}{b + c}$

Step 5.3.2

Multiply the numerator and denominator of the fraction by $2$ .

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Step 5.3.2.1

Multiply $\frac{\frac{v}{2} - b c}{b + c}$ by $\frac{2}{2}$ .

$a = \frac{2}{2} \cdot \frac{\frac{v}{2} - b c}{b + c}$

Step 5.3.2.2

Combine.

$a = \frac{2 (\frac{v}{2} - b c)}{2 (b + c)}$

Step 5.3.3

Apply the distributive property.

$a = \frac{2 \frac{v}{2} + 2 (- b c)}{2 b + 2 c}$

Step 5.3.4

Cancel the common factor of $2$ .

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Step 5.3.4.1

Cancel the common factor.

$a = \frac{2 \frac{v}{2} + 2 (- b c)}{2 b + 2 c}$

Step 5.3.4.2

Rewrite the expression.

$a = \frac{v + 2 (- b c)}{2 b + 2 c}$

Step 5.3.5

Multiply $- 1$ by $2$ .

$a = \frac{v - 2 b c}{2 b + 2 c}$

Step 5.3.6

Factor $2$ out of $2 b + 2 c$ .

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Step 5.3.6.1

Factor $2$ out of $2 b$ .

$a = \frac{v - 2 b c}{2 (b) + 2 c}$

Step 5.3.6.2

Factor $2$ out of $2 c$ .

$a = \frac{v - 2 b c}{2 (b) + 2 (c)}$

Step 5.3.6.3

Factor $2$ out of $2 (b) + 2 (c)$ .

$a = \frac{v - 2 b c}{2 (b + c)}$