Algebra Examples

$f (x) = {(x + 3)}^{2} + 4$

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 $x$ .

$0 = {(x + 3)}^{2} + 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as ${(x + 3)}^{2} + 4 = 0$ .

${(x + 3)}^{2} + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

${(x + 3)}^{2} = - 4$

Step 1.2.3

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

$x + 3 = \pm \sqrt{- 4}$

Step 1.2.4

Simplify $\pm \sqrt{- 4}$ .

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Step 1.2.4.1

Rewrite $- 4$ as $- 1 (4)$ .

$x + 3 = \pm \sqrt{- 1 (4)}$

Step 1.2.4.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x + 3 = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 1.2.4.3

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

$x + 3 = \pm i \cdot \sqrt{4}$

Step 1.2.4.4

Rewrite $4$ as $2^{2}$ .

$x + 3 = \pm i \cdot \sqrt{2^{2}}$

Step 1.2.4.5

Pull terms out from under the radical, assuming positive real numbers.

$x + 3 = \pm i \cdot 2$

Step 1.2.4.6

Move $2$ to the left of $i$ .

$x + 3 = \pm 2 i$

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.

$x + 3 = 2 i$

Step 1.2.5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$x = 2 i - 3$

Step 1.2.5.2.2

Reorder $2 i$ and $- 3$ .

$x = - 3 + 2 i$

Step 1.2.5.3

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

$x + 3 = - 2 i$

Step 1.2.5.4

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.4.1

Subtract $3$ from both sides of the equation.

$x = - 2 i - 3$

Step 1.2.5.4.2

Reorder $- 2 i$ and $- 3$ .

$x = - 3 - 2 i$

Step 1.2.5.5

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

$x = - 3 + 2 i, - 3 - 2 i$

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 $y$ .

$y = {((0) + 3)}^{2} + 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = {(0 + 3)}^{2} + 4$

Step 2.2.2

Remove parentheses.

$y = {((0) + 3)}^{2} + 4$

Step 2.2.3

Simplify ${((0) + 3)}^{2} + 4$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $3$ .

$y = 3^{2} + 4$

Step 2.2.3.1.2

Raise $3$ to the power of $2$ .

$y = 9 + 4$

Step 2.2.3.2

Add $9$ and $4$ .

$y = 13$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 13)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 13)$

Step 4