Precalculus Examples

${(x - 2)}^{2} - {(y - 1)}^{2} = 9$

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 $x$ .

${(x - 2)}^{2} - {((0) - 1)}^{2} = 9$

Step 1.2

Solve the equation.

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Step 1.2.1

Add ${((0) - 1)}^{2}$ to both sides of the equation.

${(x - 2)}^{2} = 9 + {((0) - 1)}^{2}$

Step 1.2.2

Subtract $1$ from $0$ .

${(x - 2)}^{2} = 9 + {(- 1)}^{2}$

Step 1.2.3

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

$x - 2 = \pm \sqrt{9 + {(- 1)}^{2}}$

Step 1.2.4

Simplify $\pm \sqrt{9 + {(- 1)}^{2}}$ .

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Step 1.2.4.1

Raise $- 1$ to the power of $2$ .

$x - 2 = \pm \sqrt{9 + 1}$

Step 1.2.4.2

Add $9$ and $1$ .

$x - 2 = \pm \sqrt{10}$

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.

$x - 2 = \sqrt{10}$

Step 1.2.5.2

Add $2$ to both sides of the equation.

$x = \sqrt{10} + 2$

Step 1.2.5.3

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

$x - 2 = - \sqrt{10}$

Step 1.2.5.4

Add $2$ to both sides of the equation.

$x = - \sqrt{10} + 2$

Step 1.2.5.5

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

$x = \sqrt{10} + 2, - \sqrt{10} + 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{10} + 2, 0), (- \sqrt{10} + 2, 0)$

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 $y$ .

${((0) - 2)}^{2} - {(y - 1)}^{2} = 9$

Step 2.2

Solve the equation.

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Step 2.2.1

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

$- {(y - 1)}^{2} = 9 - {((0) - 2)}^{2}$

Step 2.2.2

Simplify $9 - {((0) - 2)}^{2}$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Subtract $2$ from $0$ .

$- {(y - 1)}^{2} = 9 - {(- 2)}^{2}$

Step 2.2.2.1.2

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

$- {(y - 1)}^{2} = 9 - 1 \cdot 4$

Step 2.2.2.1.3

Multiply $- 1$ by $4$ .

$- {(y - 1)}^{2} = 9 - 4$

Step 2.2.2.2

Subtract $4$ from $9$ .

$- {(y - 1)}^{2} = 5$

Step 2.2.3

Divide each term in $- {(y - 1)}^{2} = 5$ by $- 1$ and simplify.

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Step 2.2.3.1

Divide each term in $- {(y - 1)}^{2} = 5$ by $- 1$ .

$\frac{- {(y - 1)}^{2}}{- 1} = \frac{5}{- 1}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(y - 1)}^{2}}{1} = \frac{5}{- 1}$

Step 2.2.3.2.2

Divide ${(y - 1)}^{2}$ by $1$ .

${(y - 1)}^{2} = \frac{5}{- 1}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $5$ by $- 1$ .

${(y - 1)}^{2} = - 5$

Step 2.2.4

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

$y - 1 = \pm \sqrt{- 5}$

Step 2.2.5

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

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Step 2.2.5.1

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

$y - 1 = \pm \sqrt{- 1 (5)}$

Step 2.2.5.2

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

$y - 1 = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 2.2.5.3

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

$y - 1 = \pm i \sqrt{5}$

Step 2.2.6

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

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Step 2.2.6.1

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

$y - 1 = i \sqrt{5}$

Step 2.2.6.2

Add $1$ to both sides of the equation.

$y = i \sqrt{5} + 1$

Step 2.2.6.3

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

$y - 1 = - i \sqrt{5}$

Step 2.2.6.4

Add $1$ to both sides of the equation.

$y = - i \sqrt{5} + 1$

Step 2.2.6.5

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

$y = i \sqrt{5} + 1, - i \sqrt{5} + 1$

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): $(\sqrt{10} + 2, 0), (- \sqrt{10} + 2, 0)$

y-intercept(s): $None$

Step 4