Algebra Examples

$\frac{1}{2} r^{2} - 10 = \frac{3}{2} r^{2}$

Step 1

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

$\frac{r^{2}}{2} - 10 = \frac{3}{2} r^{2}$

Step 2

Combine $\frac{3}{2}$ and $r^{2}$ .

$\frac{r^{2}}{2} - 10 = \frac{3 r^{2}}{2}$

Step 3

Move all terms containing $r$ to the left side of the equation.

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Step 3.1

Subtract $\frac{3 r^{2}}{2}$ from both sides of the equation.

$\frac{r^{2}}{2} - 10 - \frac{3 r^{2}}{2} = 0$

Step 3.2

Combine the numerators over the common denominator.

$- 10 + \frac{r^{2} - 3 r^{2}}{2} = 0$

Step 3.3

Subtract $3 r^{2}$ from $r^{2}$ .

$- 10 + \frac{- 2 r^{2}}{2} = 0$

Step 3.4

Cancel the common factor of $- 2$ and $2$ .

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Step 3.4.1

Factor $2$ out of $- 2 r^{2}$ .

$- 10 + \frac{2 (- r^{2})}{2} = 0$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $2$ .

$- 10 + \frac{2 (- r^{2})}{2 (1)} = 0$

Step 3.4.2.2

Cancel the common factor.

$- 10 + \frac{2 (- r^{2})}{2 \cdot 1} = 0$

Step 3.4.2.3

Rewrite the expression.

$- 10 + \frac{- r^{2}}{1} = 0$

Step 3.4.2.4

Divide $- r^{2}$ by $1$ .

$- 10 - r^{2} = 0$

Step 4

Add $10$ to both sides of the equation.

$- r^{2} = 10$

Step 5

Divide each term in $- r^{2} = 10$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- r^{2} = 10$ by $- 1$ .

$\frac{- r^{2}}{- 1} = \frac{10}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{r^{2}}{1} = \frac{10}{- 1}$

Step 5.2.2

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

$r^{2} = \frac{10}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $10$ by $- 1$ .

$r^{2} = - 10$

Step 6

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

$r = \pm \sqrt{- 10}$

Step 7

Simplify $\pm \sqrt{- 10}$ .

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Step 7.1

Rewrite $- 10$ as $- 1 (10)$ .

$r = \pm \sqrt{- 1 (10)}$

Step 7.2

Rewrite $\sqrt{- 1 (10)}$ as $\sqrt{- 1} \cdot \sqrt{10}$ .

$r = \pm \sqrt{- 1} \cdot \sqrt{10}$

Step 7.3

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \pm i \sqrt{10}$

Step 8

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

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Step 8.1

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

$r = i \sqrt{10}$

Step 8.2

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

$r = - i \sqrt{10}$

Step 8.3

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

$r = i \sqrt{10}, - i \sqrt{10}$