Algebra Examples

$f (x) = 5 x^{2} + 6$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 5 x^{2} + 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5 x^{2} + 6 = 0$ .

$5 x^{2} + 6 = 0$

Step 1.2.2

Subtract $6$ from both sides of the equation.

$5 x^{2} = - 6$

Step 1.2.3

Divide each term in $5 x^{2} = - 6$ by $5$ and simplify.

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Step 1.2.3.1

Divide each term in $5 x^{2} = - 6$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{- 6}{5}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $5$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{- 6}{5}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{- 6}{5}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{6}{5}$

Step 1.2.4

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

$x = \pm \sqrt{- \frac{6}{5}}$

Step 1.2.5

Simplify $\pm \sqrt{- \frac{6}{5}}$ .

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Step 1.2.5.1

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

$x = \pm \sqrt{i^{2} \frac{6}{5}}$

Step 1.2.5.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{6}{5}}$

Step 1.2.5.3

Rewrite $\sqrt{\frac{6}{5}}$ as $\frac{\sqrt{6}}{\sqrt{5}}$ .

$x = \pm i \frac{\sqrt{6}}{\sqrt{5}}$

Step 1.2.5.4

Multiply $\frac{\sqrt{6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i (\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 1.2.5.5

Combine and simplify the denominator.

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Step 1.2.5.5.1

Multiply $\frac{\sqrt{6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.2.5.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.2.5.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.2.5.5.4

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

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.2.5.5.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.2.5.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.2.5.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.2.5.5.6.2

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

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.5.6.3

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

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.2.5.5.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.5.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.2.5.5.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{5^{1}}$

Step 1.2.5.5.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{6} \sqrt{5}}{5}$

Step 1.2.5.6

Simplify the numerator.

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Step 1.2.5.6.1

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{6 \cdot 5}}{5}$

Step 1.2.5.6.2

Multiply $6$ by $5$ .

$x = \pm i \frac{\sqrt{30}}{5}$

Step 1.2.5.7

Combine $i$ and $\frac{\sqrt{30}}{5}$ .

$x = \pm \frac{i \sqrt{30}}{5}$

Step 1.2.6

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

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Step 1.2.6.1

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

$x = \frac{i \sqrt{30}}{5}$

Step 1.2.6.2

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

$x = - \frac{i \sqrt{30}}{5}$

Step 1.2.6.3

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

$x = \frac{i \sqrt{30}}{5}, - \frac{i \sqrt{30}}{5}$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 5 {(0)}^{2} + 6$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 5 \cdot 0^{2} + 6$

Step 2.2.2

Remove parentheses.

$y = 5 {(0)}^{2} + 6$

Step 2.2.3

Simplify $5 {(0)}^{2} + 6$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = 5 \cdot 0 + 6$

Step 2.2.3.1.2

Multiply $5$ by $0$ .

$y = 0 + 6$

Step 2.2.3.2

Add $0$ and $6$ .

$y = 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 6)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 6)$

Step 4