Finite Math Examples

$h (r) = 3 r^{2} + 5$

Step 1

Set $3 r^{2} + 5$ equal to $0$ .

$3 r^{2} + 5 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $5$ from both sides of the equation.

$3 r^{2} = - 5$

Step 2.2

Divide each term in $3 r^{2} = - 5$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 r^{2} = - 5$ by $3$ .

$\frac{3 r^{2}}{3} = \frac{- 5}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 r^{2}}{3} = \frac{- 5}{3}$

Step 2.2.2.1.2

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

$r^{2} = \frac{- 5}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$r^{2} = - \frac{5}{3}$

Step 2.3

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

$r = \pm \sqrt{- \frac{5}{3}}$

Step 2.4

Simplify $\pm \sqrt{- \frac{5}{3}}$ .

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Step 2.4.1

Rewrite $- 1$ as $i^{2}$ .

$r = \pm \sqrt{i^{2} \frac{5}{3}}$

Step 2.4.2

Pull terms out from under the radical.

$r = \pm i \sqrt{\frac{5}{3}}$

Step 2.4.3

Rewrite $\sqrt{\frac{5}{3}}$ as $\frac{\sqrt{5}}{\sqrt{3}}$ .

$r = \pm i \frac{\sqrt{5}}{\sqrt{3}}$

Step 2.4.4

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$r = \pm i (\frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.4.5

Combine and simplify the denominator.

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Step 2.4.5.1

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.4.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.4.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.4.5.4

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

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.4.5.5

Add $1$ and $1$ .

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.4.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.4.5.6.2

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

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.4.5.6.3

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

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.4.5.6.4

Cancel the common factor of $2$ .

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Step 2.4.5.6.4.1

Cancel the common factor.

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.4.5.6.4.2

Rewrite the expression.

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{3^{1}}$

Step 2.4.5.6.5

Evaluate the exponent.

$r = \pm i \frac{\sqrt{5} \sqrt{3}}{3}$

Step 2.4.6

Simplify the numerator.

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Step 2.4.6.1

Combine using the product rule for radicals.

$r = \pm i \frac{\sqrt{5 \cdot 3}}{3}$

Step 2.4.6.2

Multiply $5$ by $3$ .

$r = \pm i \frac{\sqrt{15}}{3}$

Step 2.4.7

Combine $i$ and $\frac{\sqrt{15}}{3}$ .

$r = \pm \frac{i \sqrt{15}}{3}$

Step 2.5

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

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Step 2.5.1

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

$r = \frac{i \sqrt{15}}{3}$

Step 2.5.2

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

$r = - \frac{i \sqrt{15}}{3}$

Step 2.5.3

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

$r = \frac{i \sqrt{15}}{3}, - \frac{i \sqrt{15}}{3}$

Step 3