Precalculus Examples

$f (x) = {(x - 4)}^{2} + 7$

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 $x$ .

$0 = {(x - 4)}^{2} + 7$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as ${(x - 4)}^{2} + 7 = 0$ .

${(x - 4)}^{2} + 7 = 0$

Step 1.2.2

Subtract $7$ from both sides of the equation.

${(x - 4)}^{2} = - 7$

Step 1.2.3

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

$x - 4 = \pm \sqrt{- 7}$

Step 1.2.4

Simplify $\pm \sqrt{- 7}$ .

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Step 1.2.4.1

Rewrite $- 7$ as $- 1 (7)$ .

$x - 4 = \pm \sqrt{- 1 (7)}$

Step 1.2.4.2

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x - 4 = \pm \sqrt{- 1} \cdot \sqrt{7}$

Step 1.2.4.3

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

$x - 4 = \pm i \sqrt{7}$

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.

$x - 4 = i \sqrt{7}$

Step 1.2.5.2

Add $4$ to both sides of the equation.

$x = i \sqrt{7} + 4$

Step 1.2.5.3

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

$x - 4 = - i \sqrt{7}$

Step 1.2.5.4

Add $4$ to both sides of the equation.

$x = - i \sqrt{7} + 4$

Step 1.2.5.5

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

$x = i \sqrt{7} + 4, - i \sqrt{7} + 4$

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 $y$ .

$y = {((0) - 4)}^{2} + 7$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = {(0 - 4)}^{2} + 7$

Step 2.2.2

Remove parentheses.

$y = {((0) - 4)}^{2} + 7$

Step 2.2.3

Simplify ${((0) - 4)}^{2} + 7$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Subtract $4$ from $0$ .

$y = {(- 4)}^{2} + 7$

Step 2.2.3.1.2

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

$y = 16 + 7$

Step 2.2.3.2

Add $16$ and $7$ .

$y = 23$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 23)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 23)$

Step 4