Precalculus Examples

$\frac{x^{2}}{9} + \frac{y^{2}}{25} = 1$

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 $x$ .

$\frac{x^{2}}{9} + \frac{{(0)}^{2}}{25} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $\frac{x^{2}}{9} + \frac{{(0)}^{2}}{25}$ .

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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 $0$ .

$\frac{x^{2}}{9} + \frac{0}{25} = 1$

Step 1.2.1.1.2

Divide $0$ by $25$ .

$\frac{x^{2}}{9} + 0 = 1$

Step 1.2.1.2

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

$\frac{x^{2}}{9} = 1$

Step 1.2.2

Multiply both sides of the equation by $9$ .

$9 \frac{x^{2}}{9} = 9 \cdot 1$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Cancel the common factor of $9$ .

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Step 1.2.3.1.1.1

Cancel the common factor.

$9 \frac{x^{2}}{9} = 9 \cdot 1$

Step 1.2.3.1.1.2

Rewrite the expression.

$x^{2} = 9 \cdot 1$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $9$ by $1$ .

$x^{2} = 9$

Step 1.2.4

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

$x = \pm \sqrt{9}$

Step 1.2.5

Simplify $\pm \sqrt{9}$ .

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Step 1.2.5.1

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

$x = \pm \sqrt{3^{2}}$

Step 1.2.5.2

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

$x = \pm 3$

Step 1.2.6

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

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Step 1.2.6.1

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

$x = 3$

Step 1.2.6.2

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

$x = - 3$

Step 1.2.6.3

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

$x = 3, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 3, 0)$

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 $y$ .

$\frac{{(0)}^{2}}{9} + \frac{y^{2}}{25} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $\frac{{(0)}^{2}}{9} + \frac{y^{2}}{25}$ .

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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 $0$ .

$\frac{0}{9} + \frac{y^{2}}{25} = 1$

Step 2.2.1.1.2

Divide $0$ by $9$ .

$0 + \frac{y^{2}}{25} = 1$

Step 2.2.1.2

Add $0$ and $\frac{y^{2}}{25}$ .

$\frac{y^{2}}{25} = 1$

Step 2.2.2

Multiply both sides of the equation by $25$ .

$25 \frac{y^{2}}{25} = 25 \cdot 1$

Step 2.2.3

Simplify both sides of the equation.

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Step 2.2.3.1

Simplify the left side.

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Step 2.2.3.1.1

Cancel the common factor of $25$ .

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Step 2.2.3.1.1.1

Cancel the common factor.

$25 \frac{y^{2}}{25} = 25 \cdot 1$

Step 2.2.3.1.1.2

Rewrite the expression.

$y^{2} = 25 \cdot 1$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

Multiply $25$ by $1$ .

$y^{2} = 25$

Step 2.2.4

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

$y = \pm \sqrt{25}$

Step 2.2.5

Simplify $\pm \sqrt{25}$ .

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

$y = \pm 5$

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.

$y = 5$

Step 2.2.6.2

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

$y = - 5$

Step 2.2.6.3

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

$y = 5, - 5$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 3, 0)$

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

Step 4