Find the x and y Intercepts f(x)=(2x+4)^2+5

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

Find the x-intercepts.

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

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

Solve the equation.

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

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

Subtract $5$ from both sides of the equation.

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

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

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

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

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

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

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

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

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

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

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

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

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

Subtract $4$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $2$ .

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

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

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

Subtract $4$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Divide $- 4$ by $2$ .

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

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

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

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

x-intercept(s): $None$

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

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

Simplify each term.

Multiply $2$ by $0$ .

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

Add $0$ and $4$ .

$y = 4^{2} + 5$

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

$y = 16 + 5$

Add $16$ and $5$ .

$y = 21$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $None$

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

Find the x and y Intercepts f(x)=(2x+4)^2+5

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