Pre-Algebra Examples

$B = \frac{1}{2} \cdot (f (s + z) f o r s)$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (f (s + z) f o r s) = B$ .

$\frac{1}{2} \cdot (f (s + z) f o r s) = B$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (f (s + z) f o r s)) = 2 B$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 (\frac{1}{2} \cdot (f (s + z) f o r s))$ .

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Step 3.1.1

Multiply $f$ by $f$ by adding the exponents.

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Step 3.1.1.1

Move $f$ .

$2 (\frac{1}{2} \cdot (f \cdot f (s + z) o r s)) = 2 B$

Step 3.1.1.2

Multiply $f$ by $f$ .

$2 (\frac{1}{2} \cdot (f^{2} (s + z) o r s)) = 2 B$

Step 3.1.2

Apply the distributive property.

$2 (\frac{1}{2} \cdot ((f^{2} s + f^{2} z) o r s)) = 2 B$

Step 3.1.3

Apply the distributive property.

$2 (\frac{1}{2} \cdot ((f^{2} s o + f^{2} z o) r s)) = 2 B$

Step 3.1.4

Apply the distributive property.

$2 (\frac{1}{2} \cdot ((f^{2} s o r + f^{2} z o r) s)) = 2 B$

Step 3.1.5

Apply the distributive property.

$2 (\frac{1}{2} \cdot (f^{2} s o r s + f^{2} z o r s)) = 2 B$

Step 3.1.6

Multiply $s$ by $s$ by adding the exponents.

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Step 3.1.6.1

Move $s$ .

$2 (\frac{1}{2} \cdot (f^{2} (s \cdot s) o r + f^{2} z o r s)) = 2 B$

Step 3.1.6.2

Multiply $s$ by $s$ .

$2 (\frac{1}{2} \cdot (f^{2} s^{2} o r + f^{2} z o r s)) = 2 B$

Step 3.1.7

Apply the distributive property.

$2 (\frac{1}{2} (f^{2} s^{2} o r) + \frac{1}{2} (f^{2} z o r s)) = 2 B$

Step 3.1.8

Multiply $\frac{1}{2} (f^{2} s^{2} o r)$ .

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Step 3.1.8.1

Combine $f^{2}$ and $\frac{1}{2}$ .

$2 (\frac{f^{2}}{2} (s^{2} o r) + \frac{1}{2} (f^{2} z o r s)) = 2 B$

Step 3.1.8.2

Combine $s^{2}$ and $\frac{f^{2}}{2}$ .

$2 (\frac{s^{2} f^{2}}{2} (o r) + \frac{1}{2} (f^{2} z o r s)) = 2 B$

Step 3.1.8.3

Combine $o$ and $\frac{s^{2} f^{2}}{2}$ .

$2 (\frac{o (s^{2} f^{2})}{2} r + \frac{1}{2} (f^{2} z o r s)) = 2 B$

Step 3.1.8.4

Combine $\frac{o (s^{2} f^{2})}{2}$ and $r$ .

$2 (\frac{o (s^{2} f^{2}) r}{2} + \frac{1}{2} (f^{2} z o r s)) = 2 B$

Step 3.1.9

Multiply $\frac{1}{2} (f^{2} z o r s)$ .

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Step 3.1.9.1

Combine $f^{2}$ and $\frac{1}{2}$ .

$2 (\frac{o (s^{2} f^{2}) r}{2} + \frac{f^{2}}{2} (z o r s)) = 2 B$

Step 3.1.9.2

Combine $z$ and $\frac{f^{2}}{2}$ .

$2 (\frac{o (s^{2} f^{2}) r}{2} + \frac{z f^{2}}{2} (o r s)) = 2 B$

Step 3.1.9.3

Combine $o$ and $\frac{z f^{2}}{2}$ .

$2 (\frac{o (s^{2} f^{2}) r}{2} + \frac{o (z f^{2})}{2} (r s)) = 2 B$

Step 3.1.9.4

Combine $r$ and $\frac{o (z f^{2})}{2}$ .

$2 (\frac{o (s^{2} f^{2}) r}{2} + \frac{r (o (z f^{2}))}{2} s) = 2 B$

Step 3.1.9.5

Combine $\frac{r (o (z f^{2}))}{2}$ and $s$ .

$2 (\frac{o (s^{2} f^{2}) r}{2} + \frac{r (o (z f^{2})) s}{2}) = 2 B$

Step 3.1.10

Remove parentheses.

$2 (\frac{o s^{2} f^{2} r}{2} + \frac{r o z f^{2} s}{2}) = 2 B$

Step 3.1.11

Simplify terms.

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Step 3.1.11.1

Apply the distributive property.

$2 \frac{o s^{2} f^{2} r}{2} + 2 \frac{r o z f^{2} s}{2} = 2 B$

Step 3.1.11.2

Cancel the common factor of $2$ .

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Step 3.1.11.2.1

Cancel the common factor.

$2 \frac{o s^{2} f^{2} r}{2} + 2 \frac{r o z f^{2} s}{2} = 2 B$

Step 3.1.11.2.2

Rewrite the expression.

$o s^{2} f^{2} r + 2 \frac{r o z f^{2} s}{2} = 2 B$

Step 3.1.11.3

Cancel the common factor of $2$ .

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Step 3.1.11.3.1

Cancel the common factor.

$o s^{2} f^{2} r + 2 \frac{r o z f^{2} s}{2} = 2 B$

Step 3.1.11.3.2

Rewrite the expression.

$o s^{2} f^{2} r + r o z f^{2} s = 2 B$

Step 4

Subtract $2 B$ from both sides of the equation.

$o s^{2} f^{2} r + r o z f^{2} s - 2 B = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = o f^{2} r$ , $b = r o z f^{2}$ , and $c = - 2 B$ into the quadratic formula and solve for $s$ .

$\frac{- r o z f^{2} \pm \sqrt{{(r o z f^{2})}^{2} - 4 \cdot (o f^{2} r \cdot (- 2 B))}}{2 (o f^{2} r)}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1.1.1

Apply the product rule to $r o z f^{2}$ .

$s = \frac{- r o z f^{2} \pm \sqrt{{(r o z)}^{2} {(f^{2})}^{2} - 4 \cdot (o f^{2} r) \cdot (- 2 B)}}{2 o f^{2} r}$

Step 7.1.1.2

Apply the product rule to $r o z$ .

$s = \frac{- r o z f^{2} \pm \sqrt{{(r o)}^{2} z^{2} {(f^{2})}^{2} - 4 \cdot (o f^{2} r) \cdot (- 2 B)}}{2 o f^{2} r}$

Step 7.1.1.3

Apply the product rule to $r o$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r^{2} o^{2} z^{2} {(f^{2})}^{2} - 4 \cdot (o f^{2} r) \cdot (- 2 B)}}{2 o f^{2} r}$

Step 7.1.2

Multiply the exponents in ${(f^{2})}^{2}$ .

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Step 7.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r^{2} o^{2} z^{2} f^{2 \cdot 2} - 4 \cdot (o f^{2} r) \cdot (- 2 B)}}{2 o f^{2} r}$

Step 7.1.2.2

Multiply $2$ by $2$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r^{2} o^{2} z^{2} f^{4} - 4 \cdot (o f^{2} r) \cdot (- 2 B)}}{2 o f^{2} r}$

Step 7.1.3

Multiply $- 2$ by $- 4$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r^{2} o^{2} z^{2} f^{4} + 8 o f^{2} r B}}{2 o f^{2} r}$

Step 7.1.4

Factor $r o f^{2}$ out of $r^{2} o^{2} z^{2} f^{4} + 8 o f^{2} r B$ .

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Step 7.1.4.1

Factor $r o f^{2}$ out of $r^{2} o^{2} z^{2} f^{4}$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r o f^{2} (r o z^{2} f^{2}) + 8 o f^{2} r B}}{2 o f^{2} r}$

Step 7.1.4.2

Factor $r o f^{2}$ out of $8 o f^{2} r B$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r o f^{2} (r o z^{2} f^{2}) + r o f^{2} (8 B)}}{2 o f^{2} r}$

Step 7.1.4.3

Factor $r o f^{2}$ out of $r o f^{2} (r o z^{2} f^{2}) + r o f^{2} (8 B)$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r o f^{2} (r o z^{2} f^{2} + 8 B)}}{2 o f^{2} r}$

Step 7.1.5

Rewrite $r o f^{2} (r o z^{2} f^{2} + 8 B)$ as $f^{2} (r o (r o z^{2} f^{2} + 8 B))$ .

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Step 7.1.5.1

Move $o$ .

$s = \frac{- r o z f^{2} \pm \sqrt{r f^{2} o (r o z^{2} f^{2} + 8 B)}}{2 o f^{2} r}$

Step 7.1.5.2

Reorder $r$ and $f^{2}$ .

$s = \frac{- r o z f^{2} \pm \sqrt{f^{2} r o (r o z^{2} f^{2} + 8 B)}}{2 o f^{2} r}$

Step 7.1.5.3

Add parentheses.

$s = \frac{- r o z f^{2} \pm \sqrt{f^{2} r (o (r o z^{2} f^{2} + 8 B))}}{2 o f^{2} r}$

Step 7.1.5.4

Add parentheses.

$s = \frac{- r o z f^{2} \pm \sqrt{f^{2} (r o (r o z^{2} f^{2} + 8 B))}}{2 o f^{2} r}$

Step 7.1.6

Pull terms out from under the radical.

$s = \frac{- r o z f^{2} \pm f \sqrt{r o (r o z^{2} f^{2} + 8 B)}}{2 o f^{2} r}$

Step 7.2

Simplify $\frac{- r o z f^{2} \pm f \sqrt{r o (r o z^{2} f^{2} + 8 B)}}{2 o f^{2} r}$ .

$s = \frac{- r o z f \pm \sqrt{r o (r o z^{2} f^{2} + 8 B)}}{2 o f r}$

Step 8

The final answer is the combination of both solutions.

$s = - \frac{r o z f - \sqrt{r o (r o z^{2} f^{2} + 8 B)}}{2 o f r}$

$s = - \frac{r o z f + \sqrt{r o (r o z^{2} f^{2} + 8 B)}}{2 o f r}$