Finite Math Examples

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

Step 1

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

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

Step 2

Multiply both sides by $1 - r$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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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.

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

Step 3.1.1.1.2

Rewrite the expression.

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

Step 3.1.1.2

Reorder $1$ and $- r x^{2}$ .

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

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.

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

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Multiply $7$ by $1$ .

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

Step 3.2.1.2.2

Multiply $- 1$ by $7$ .

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

Step 3.2.1.2.3

Reorder $7$ and $- 7 r$ .

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

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 $1$ from both sides of the equation.

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

Step 4.1.2

Subtract $1$ from $7$ .

$- r x^{2} = - 7 r + 6$

Step 4.2

Divide each term in $- r x^{2} = - 7 r + 6$ by $- r$ and simplify.

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Step 4.2.1

Divide each term in $- r x^{2} = - 7 r + 6$ by $- r$ .

$\frac{- r x^{2}}{- r} = \frac{- 7 r}{- r} + \frac{6}{- r}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{r x^{2}}{r} = \frac{- 7 r}{- r} + \frac{6}{- r}$

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.

$\frac{r x^{2}}{r} = \frac{- 7 r}{- r} + \frac{6}{- r}$

Step 4.2.2.2.2

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

$x^{2} = \frac{- 7 r}{- r} + \frac{6}{- r}$

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.

$x^{2} = \frac{- 7 r}{- r} + \frac{6}{- r}$

Step 4.2.3.1.1.2

Rewrite the expression.

$x^{2} = \frac{- 7}{- 1} + \frac{6}{- r}$

Step 4.2.3.1.1.3

Move the negative one from the denominator of $\frac{- 7}{- 1}$ .

$x^{2} = - 1 \cdot - 7 + \frac{6}{- r}$

Step 4.2.3.1.2

Rewrite $- 1 \cdot - 7$ as $- - 7$ .

$x^{2} = - - 7 + \frac{6}{- r}$

Step 4.2.3.1.3

Multiply $- 1$ by $- 7$ .

$x^{2} = 7 + \frac{6}{- r}$

Step 4.2.3.1.4

Move the negative in front of the fraction.

$x^{2} = 7 - \frac{6}{r}$

Step 4.3

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

$x = \pm \sqrt{7 - \frac{6}{r}}$

Step 4.4

Simplify $\pm \sqrt{7 - \frac{6}{r}}$ .

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Step 4.4.1

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

$x = \pm \sqrt{\frac{7 r}{r} - \frac{6}{r}}$

Step 4.4.2

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{7 r - 6}{r}}$

Step 4.4.3

Rewrite $\sqrt{\frac{7 r - 6}{r}}$ as $\frac{\sqrt{7 r - 6}}{\sqrt{r}}$ .

$x = \pm \frac{\sqrt{7 r - 6}}{\sqrt{r}}$

Step 4.4.4

Multiply $\frac{\sqrt{7 r - 6}}{\sqrt{r}}$ by $\frac{\sqrt{r}}{\sqrt{r}}$ .

$x = \pm \frac{\sqrt{7 r - 6}}{\sqrt{r}} \cdot \frac{\sqrt{r}}{\sqrt{r}}$

Step 4.4.5

Combine and simplify the denominator.

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Step 4.4.5.1

Multiply $\frac{\sqrt{7 r - 6}}{\sqrt{r}}$ by $\frac{\sqrt{r}}{\sqrt{r}}$ .

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{\sqrt{r} \sqrt{r}}$

Step 4.4.5.2

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

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{{\sqrt{r}}^{1} \sqrt{r}}$

Step 4.4.5.3

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

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{{\sqrt{r}}^{1} {\sqrt{r}}^{1}}$

Step 4.4.5.4

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

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{{\sqrt{r}}^{1 + 1}}$

Step 4.4.5.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{{\sqrt{r}}^{2}}$

Step 4.4.5.6

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

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Step 4.4.5.6.1

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

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{{(r^{\frac{1}{2}})}^{2}}$

Step 4.4.5.6.2

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

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{r^{\frac{1}{2} \cdot 2}}$

Step 4.4.5.6.3

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

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{r^{\frac{2}{2}}}$

Step 4.4.5.6.4

Cancel the common factor of $2$ .

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Step 4.4.5.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{r^{\frac{2}{2}}}$

Step 4.4.5.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{r^{1}}$

Step 4.4.5.6.5

Simplify.

$x = \pm \frac{\sqrt{7 r - 6} \sqrt{r}}{r}$

Step 4.4.6

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(7 r - 6) r}}{r}$

Step 4.4.7

Reorder factors in $\pm \frac{\sqrt{(7 r - 6) r}}{r}$ .

$x = \pm \frac{\sqrt{r (7 r - 6)}}{r}$

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.

$x = \frac{\sqrt{r (7 r - 6)}}{r}$

Step 4.5.2

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

$x = - \frac{\sqrt{r (7 r - 6)}}{r}$

Step 4.5.3

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

$x = \frac{\sqrt{r (7 r - 6)}}{r}$

$x = - \frac{\sqrt{r (7 r - 6)}}{r}$

$x = \frac{\sqrt{r (7 r - 6)}}{r}$

$x = - \frac{\sqrt{r (7 r - 6)}}{r}$

$x = \frac{\sqrt{r (7 r - 6)}}{r}$

$x = - \frac{\sqrt{r (7 r - 6)}}{r}$