Algebra Examples

$f (x) = {(2 x + 4)}^{2} + 5$

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 $x$ .

$0 = {(2 x + 4)}^{2} + 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as ${(2 x + 4)}^{2} + 5 = 0$ .

${(2 x + 4)}^{2} + 5 = 0$

Step 1.2.2

Subtract $5$ from both sides of the equation.

${(2 x + 4)}^{2} = - 5$

Step 1.2.3

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

$2 x + 4 = \pm \sqrt{- 5}$

Step 1.2.4

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

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Step 1.2.4.1

Rewrite $- 5$ as $- 1 (5)$ .

$2 x + 4 = \pm \sqrt{- 1 (5)}$

Step 1.2.4.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$2 x + 4 = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 1.2.4.3

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

$2 x + 4 = \pm i \sqrt{5}$

Step 1.2.5

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

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Step 1.2.5.1

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

$2 x + 4 = i \sqrt{5}$

Step 1.2.5.2

Subtract $4$ from both sides of the equation.

$2 x = i \sqrt{5} - 4$

Step 1.2.5.3

Divide each term in $2 x = i \sqrt{5} - 4$ by $2$ and simplify.

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Step 1.2.5.3.1

Divide each term in $2 x = i \sqrt{5} - 4$ by $2$ .

$\frac{2 x}{2} = \frac{i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.3.2

Simplify the left side.

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Step 1.2.5.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.3.3

Simplify the right side.

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Step 1.2.5.3.3.1

Divide $- 4$ by $2$ .

$x = \frac{i \sqrt{5}}{2} - 2$

Step 1.2.5.4

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

$2 x + 4 = - i \sqrt{5}$

Step 1.2.5.5

Subtract $4$ from both sides of the equation.

$2 x = - i \sqrt{5} - 4$

Step 1.2.5.6

Divide each term in $2 x = - i \sqrt{5} - 4$ by $2$ and simplify.

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Step 1.2.5.6.1

Divide each term in $2 x = - i \sqrt{5} - 4$ by $2$ .

$\frac{2 x}{2} = \frac{- i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.6.2

Simplify the left side.

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Step 1.2.5.6.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.6.3

Simplify the right side.

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Step 1.2.5.6.3.1

Simplify each term.

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Step 1.2.5.6.3.1.1

Move the negative in front of the fraction.

$x = - \frac{i \sqrt{5}}{2} + \frac{- 4}{2}$

Step 1.2.5.6.3.1.2

Divide $- 4$ by $2$ .

$x = - \frac{i \sqrt{5}}{2} - 2$

Step 1.2.5.7

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

$x = \frac{i \sqrt{5}}{2} - 2, - \frac{i \sqrt{5}}{2} - 2$

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 $y$ .

$y = {(2 (0) + 4)}^{2} + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = {(2 (0) + 4)}^{2} + 5$

Step 2.2.2

Simplify ${(2 (0) + 4)}^{2} + 5$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $2$ by $0$ .

$y = {(0 + 4)}^{2} + 5$

Step 2.2.2.1.2

Add $0$ and $4$ .

$y = 4^{2} + 5$

Step 2.2.2.1.3

Raise $4$ to the power of $2$ .

$y = 16 + 5$

Step 2.2.2.2

Add $16$ and $5$ .

$y = 21$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 21)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 21)$

Step 4