Finite Math Examples

$7 = \frac{4 (1 - r x^{2})}{1 - r}$

Step 1

Rewrite the equation as $\frac{4 (1 - r x^{2})}{1 - r} = 7$ .

$\frac{4 (1 - r x^{2})}{1 - r} = 7$

Step 2

Multiply both sides by $1 - r$ .

$\frac{4 (1 - r x^{2})}{1 - r} (1 - r) = 7 (1 - r)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{4 (1 - r x^{2})}{1 - r} (1 - r)$ .

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Step 3.1.1.1

Cancel the common factor of $1 - r$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{4 (1 - r x^{2})}{1 - r} (1 - r) = 7 (1 - r)$

Step 3.1.1.1.2

Rewrite the expression.

$4 (1 - r x^{2}) = 7 (1 - r)$

Step 3.1.1.2

Apply the distributive property.

$4 \cdot 1 + 4 (- r x^{2}) = 7 (1 - r)$

Step 3.1.1.3

Simplify the expression.

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Step 3.1.1.3.1

Multiply $4$ by $1$ .

$4 + 4 (- r x^{2}) = 7 (1 - r)$

Step 3.1.1.3.2

Multiply $- 1$ by $4$ .

$4 - 4 (r x^{2}) = 7 (1 - r)$

Step 3.1.1.3.3

Reorder $4$ and $- 4 r x^{2}$ .

$- 4 r x^{2} + 4 = 7 (1 - r)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $7 (1 - r)$ .

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Step 3.2.1.1

Apply the distributive property.

$- 4 r x^{2} + 4 = 7 \cdot 1 + 7 (- r)$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Multiply $7$ by $1$ .

$- 4 r x^{2} + 4 = 7 + 7 (- r)$

Step 3.2.1.2.2

Multiply $- 1$ by $7$ .

$- 4 r x^{2} + 4 = 7 - 7 r$

Step 3.2.1.2.3

Reorder $7$ and $- 7 r$ .

$- 4 r x^{2} + 4 = - 7 r + 7$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1.1

Subtract $4$ from both sides of the equation.

$- 4 r x^{2} = - 7 r + 7 - 4$

Step 4.1.2

Subtract $4$ from $7$ .

$- 4 r x^{2} = - 7 r + 3$

Step 4.2

Divide each term in $- 4 r x^{2} = - 7 r + 3$ by $- 4 r$ and simplify.

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Step 4.2.1

Divide each term in $- 4 r x^{2} = - 7 r + 3$ by $- 4 r$ .

$\frac{- 4 r x^{2}}{- 4 r} = \frac{- 7 r}{- 4 r} + \frac{3}{- 4 r}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 4 r x^{2}}{- 4 r} = \frac{- 7 r}{- 4 r} + \frac{3}{- 4 r}$

Step 4.2.2.1.2

Rewrite the expression.

$\frac{r x^{2}}{r} = \frac{- 7 r}{- 4 r} + \frac{3}{- 4 r}$

Step 4.2.2.2

Cancel the common factor of $r$ .

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Step 4.2.2.2.1

Cancel the common factor.

$\frac{r x^{2}}{r} = \frac{- 7 r}{- 4 r} + \frac{3}{- 4 r}$

Step 4.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 7 r}{- 4 r} + \frac{3}{- 4 r}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $r$ .

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Step 4.2.3.1.1.1

Cancel the common factor.

$x^{2} = \frac{- 7 r}{- 4 r} + \frac{3}{- 4 r}$

Step 4.2.3.1.1.2

Rewrite the expression.

$x^{2} = \frac{- 7}{- 4} + \frac{3}{- 4 r}$

Step 4.2.3.1.2

Dividing two negative values results in a positive value.

$x^{2} = \frac{7}{4} + \frac{3}{- 4 r}$

Step 4.2.3.1.3

Move the negative in front of the fraction.

$x^{2} = \frac{7}{4} - \frac{3}{4 r}$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{7}{4} - \frac{3}{4 r}}$

Step 4.4

Simplify $\pm \sqrt{\frac{7}{4} - \frac{3}{4 r}}$ .

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Step 4.4.1

To write $\frac{7}{4}$ as a fraction with a common denominator, multiply by $\frac{r}{r}$ .

$x = \pm \sqrt{\frac{7}{4} \cdot \frac{r}{r} - \frac{3}{4 r}}$

Step 4.4.2

Multiply $\frac{7}{4}$ by $\frac{r}{r}$ .

$x = \pm \sqrt{\frac{7 r}{4 r} - \frac{3}{4 r}}$

Step 4.4.3

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{7 r - 3}{4 r}}$

Step 4.4.4

Rewrite $\frac{7 r - 3}{4 r}$ as ${(\frac{1}{2})}^{2} \frac{7 r - 3}{r}$ .

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Step 4.4.4.1

Factor the perfect power $1^{2}$ out of $7 r - 3$ .

$x = \pm \sqrt{\frac{1^{2} (7 r - 3)}{4 r}}$

Step 4.4.4.2

Factor the perfect power $2^{2}$ out of $4 r$ .

$x = \pm \sqrt{\frac{1^{2} (7 r - 3)}{2^{2} r}}$

Step 4.4.4.3

Rearrange the fraction $\frac{1^{2} (7 r - 3)}{2^{2} r}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} \frac{7 r - 3}{r}}$

Step 4.4.5

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \sqrt{\frac{7 r - 3}{r}}$

Step 4.4.6

Rewrite $\sqrt{\frac{7 r - 3}{r}}$ as $\frac{\sqrt{7 r - 3}}{\sqrt{r}}$ .

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3}}{\sqrt{r}}$

Step 4.4.7

Multiply $\frac{\sqrt{7 r - 3}}{\sqrt{r}}$ by $\frac{\sqrt{r}}{\sqrt{r}}$ .

$x = \pm \frac{1}{2} (\frac{\sqrt{7 r - 3}}{\sqrt{r}} \cdot \frac{\sqrt{r}}{\sqrt{r}})$

Step 4.4.8

Combine and simplify the denominator.

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Step 4.4.8.1

Multiply $\frac{\sqrt{7 r - 3}}{\sqrt{r}}$ by $\frac{\sqrt{r}}{\sqrt{r}}$ .

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{\sqrt{r} \sqrt{r}}$

Step 4.4.8.2

Raise $\sqrt{r}$ to the power of $1$ .

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{{\sqrt{r}}^{1} \sqrt{r}}$

Step 4.4.8.3

Raise $\sqrt{r}$ to the power of $1$ .

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{{\sqrt{r}}^{1} {\sqrt{r}}^{1}}$

Step 4.4.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{{\sqrt{r}}^{1 + 1}}$

Step 4.4.8.5

Add $1$ and $1$ .

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{{\sqrt{r}}^{2}}$

Step 4.4.8.6

Rewrite ${\sqrt{r}}^{2}$ as $r$ .

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Step 4.4.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{r}$ as $r^{\frac{1}{2}}$ .

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{{(r^{\frac{1}{2}})}^{2}}$

Step 4.4.8.6.2

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

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{r^{\frac{1}{2} \cdot 2}}$

Step 4.4.8.6.3

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

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{r^{\frac{2}{2}}}$

Step 4.4.8.6.4

Cancel the common factor of $2$ .

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Step 4.4.8.6.4.1

Cancel the common factor.

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{r^{\frac{2}{2}}}$

Step 4.4.8.6.4.2

Rewrite the expression.

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{r^{1}}$

Step 4.4.8.6.5

Simplify.

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{7 r - 3} \sqrt{r}}{r}$

Step 4.4.9

Combine using the product rule for radicals.

$x = \pm \frac{1}{2} \cdot \frac{\sqrt{(7 r - 3) r}}{r}$

Step 4.4.10

Multiply $\frac{1}{2}$ by $\frac{\sqrt{(7 r - 3) r}}{r}$ .

$x = \pm \frac{\sqrt{(7 r - 3) r}}{2 r}$

Step 4.4.11

Reorder factors in $\pm \frac{\sqrt{(7 r - 3) r}}{2 r}$ .

$x = \pm \frac{\sqrt{r (7 r - 3)}}{2 r}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{r (7 r - 3)}}{2 r}$

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{r (7 r - 3)}}{2 r}$

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{r (7 r - 3)}}{2 r}$

$x = - \frac{\sqrt{r (7 r - 3)}}{2 r}$

$x = \frac{\sqrt{r (7 r - 3)}}{2 r}$

$x = - \frac{\sqrt{r (7 r - 3)}}{2 r}$

$x = \frac{\sqrt{r (7 r - 3)}}{2 r}$

$x = - \frac{\sqrt{r (7 r - 3)}}{2 r}$