Precalculus Examples

$G (x) = {(x - 3)}^{2} + 1$

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 $x$ .

$0 = {(x - 3)}^{2} + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as ${(x - 3)}^{2} + 1 = 0$ .

${(x - 3)}^{2} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

${(x - 3)}^{2} = - 1$

Step 1.2.3

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

$x - 3 = \pm \sqrt{- 1}$

Step 1.2.4

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

$x - 3 = \pm i$

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.

$x - 3 = i$

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

Add $3$ to both sides of the equation.

$x = i + 3$

Step 1.2.5.2.2

Reorder $i$ and $3$ .

$x = 3 + i$

Step 1.2.5.3

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

$x - 3 = - i$

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

Add $3$ to both sides of the equation.

$x = - i + 3$

Step 1.2.5.4.2

Reorder $- i$ and $3$ .

$x = 3 - i$

Step 1.2.5.5

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

$x = 3 + i, 3 - i$

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 $y$ .

$y = {((0) - 3)}^{2} + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = {(0 - 3)}^{2} + 1$

Step 2.2.2

Remove parentheses.

$y = {((0) - 3)}^{2} + 1$

Step 2.2.3

Simplify ${((0) - 3)}^{2} + 1$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Subtract $3$ from $0$ .

$y = {(- 3)}^{2} + 1$

Step 2.2.3.1.2

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

$y = 9 + 1$

Step 2.2.3.2

Add $9$ and $1$ .

$y = 10$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 10)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 10)$

Step 4