Precalculus Examples

${(x + 7)}^{2} + {(y - 0)}^{2} = 25$

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 $x$ .

${(x + 7)}^{2} + {((0) - 0)}^{2} = 25$

Step 1.2

Solve the equation.

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Step 1.2.1

Subtract ${((0) - 0)}^{2}$ from both sides of the equation.

${(x + 7)}^{2} = 25 - {((0) - 0)}^{2}$

Step 1.2.2

Multiply $- 1$ by $0$ .

${(x + 7)}^{2} = 25 - {(0 + 0)}^{2}$

Step 1.2.3

Add $0$ and $0$ .

${(x + 7)}^{2} = 25 - 0^{2}$

Step 1.2.4

Raising $0$ to any positive power yields $0$ .

${(x + 7)}^{2} = 25 - 0$

Step 1.2.5

Multiply $- 1$ by $0$ .

${(x + 7)}^{2} = 25 + 0$

Step 1.2.6

Add $25$ and $0$ .

${(x + 7)}^{2} = 25$

Step 1.2.7

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

$x + 7 = \pm \sqrt{25}$

Step 1.2.8

Simplify $\pm \sqrt{25}$ .

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Step 1.2.8.1

Rewrite $25$ as $5^{2}$ .

$x + 7 = \pm \sqrt{5^{2}}$

Step 1.2.8.2

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

$x + 7 = \pm 5$

Step 1.2.9

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

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Step 1.2.9.1

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

$x + 7 = 5$

Step 1.2.9.2

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

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Step 1.2.9.2.1

Subtract $7$ from both sides of the equation.

$x = 5 - 7$

Step 1.2.9.2.2

Subtract $7$ from $5$ .

$x = - 2$

Step 1.2.9.3

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

$x + 7 = - 5$

Step 1.2.9.4

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

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Step 1.2.9.4.1

Subtract $7$ from both sides of the equation.

$x = - 5 - 7$

Step 1.2.9.4.2

Subtract $7$ from $- 5$ .

$x = - 12$

Step 1.2.9.5

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

$x = - 2, - 12$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (- 12, 0)$

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 $y$ .

${((0) + 7)}^{2} + {(y - 0)}^{2} = 25$

Step 2.2

Solve the equation.

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Step 2.2.1

Subtract ${((0) + 7)}^{2}$ from both sides of the equation.

${(y - 0)}^{2} = 25 - {((0) + 7)}^{2}$

Step 2.2.2

Multiply $- 1$ by $0$ .

${(y + 0)}^{2} = 25 - {((0) + 7)}^{2}$

Step 2.2.3

Add $0$ and $7$ .

${(y + 0)}^{2} = 25 - 7^{2}$

Step 2.2.4

Raise $7$ to the power of $2$ .

${(y + 0)}^{2} = 25 - 1 \cdot 49$

Step 2.2.5

Multiply $- 1$ by $49$ .

${(y + 0)}^{2} = 25 - 49$

Step 2.2.6

Add $y$ and $0$ .

$y^{2} = 25 - 49$

Step 2.2.7

Subtract $49$ from $25$ .

$y^{2} = - 24$

Step 2.2.8

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

$y = \pm \sqrt{- 24}$

Step 2.2.9

Simplify $\pm \sqrt{- 24}$ .

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Step 2.2.9.1

Rewrite $- 24$ as $- 1 (24)$ .

$y = \pm \sqrt{- 1 (24)}$

Step 2.2.9.2

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{24}$

Step 2.2.9.3

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

$y = \pm i \cdot \sqrt{24}$

Step 2.2.9.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.2.9.4.1

Factor $4$ out of $24$ .

$y = \pm i \cdot \sqrt{4 (6)}$

Step 2.2.9.4.2

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

$y = \pm i \cdot \sqrt{2^{2} \cdot 6}$

Step 2.2.9.5

Pull terms out from under the radical.

$y = \pm i \cdot (2 \sqrt{6})$

Step 2.2.9.6

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

$y = \pm 2 i \sqrt{6}$

Step 2.2.10

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

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Step 2.2.10.1

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

$y = 2 i \sqrt{6}$

Step 2.2.10.2

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

$y = - 2 i \sqrt{6}$

Step 2.2.10.3

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

$y = 2 i \sqrt{6}, - 2 i \sqrt{6}$

Step 2.3

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0), (- 12, 0)$

y-intercept(s): $None$

Step 4