Precalculus Examples

$9 x^{2} - 25 y^{2} = 225$

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 $x$ .

$9 x^{2} - 25 {(0)}^{2} = 225$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $9 x^{2} - 25 {(0)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$9 x^{2} - 25 \cdot 0 = 225$

Step 1.2.1.1.2

Multiply $- 25$ by $0$ .

$9 x^{2} + 0 = 225$

Step 1.2.1.2

Add $9 x^{2}$ and $0$ .

$9 x^{2} = 225$

Step 1.2.2

Divide each term in $9 x^{2} = 225$ by $9$ and simplify.

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Step 1.2.2.1

Divide each term in $9 x^{2} = 225$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{225}{9}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{225}{9}$

Step 1.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{225}{9}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $225$ by $9$ .

$x^{2} = 25$

Step 1.2.3

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

$x = \pm \sqrt{25}$

Step 1.2.4

Simplify $\pm \sqrt{25}$ .

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

$x = \pm 5$

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.

$x = 5$

Step 1.2.5.2

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

$x = - 5$

Step 1.2.5.3

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

$x = 5, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (- 5, 0)$

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 $y$ .

$9 {(0)}^{2} - 25 y^{2} = 225$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $9 {(0)}^{2} - 25 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$9 \cdot 0 - 25 y^{2} = 225$

Step 2.2.1.1.2

Multiply $9$ by $0$ .

$0 - 25 y^{2} = 225$

Step 2.2.1.2

Subtract $25 y^{2}$ from $0$ .

$- 25 y^{2} = 225$

Step 2.2.2

Divide each term in $- 25 y^{2} = 225$ by $- 25$ and simplify.

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Step 2.2.2.1

Divide each term in $- 25 y^{2} = 225$ by $- 25$ .

$\frac{- 25 y^{2}}{- 25} = \frac{225}{- 25}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $- 25$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{- 25 y^{2}}{- 25} = \frac{225}{- 25}$

Step 2.2.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{225}{- 25}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $225$ by $- 25$ .

$y^{2} = - 9$

Step 2.2.3

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

$y = \pm \sqrt{- 9}$

Step 2.2.4

Simplify $\pm \sqrt{- 9}$ .

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Step 2.2.4.1

Rewrite $- 9$ as $- 1 (9)$ .

$y = \pm \sqrt{- 1 (9)}$

Step 2.2.4.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 2.2.4.3

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

$y = \pm i \cdot \sqrt{9}$

Step 2.2.4.4

Rewrite $9$ as $3^{2}$ .

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

Step 2.2.4.5

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

$y = \pm i \cdot 3$

Step 2.2.4.6

Move $3$ to the left of $i$ .

$y = \pm 3 i$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = 3 i$

Step 2.2.5.2

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

$y = - 3 i$

Step 2.2.5.3

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

$y = 3 i, - 3 i$

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): $(5, 0), (- 5, 0)$

y-intercept(s): $None$

Step 4