Algebra Examples

$\frac{x^{2}}{9} + \frac{{(y - 3)}^{2}}{25} = 1$

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 $x$ .

$\frac{x^{2}}{9} + \frac{{((0) - 3)}^{2}}{25} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $\frac{x^{2}}{9} + \frac{{((0) - 3)}^{2}}{25}$ .

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Step 1.2.1.1

Subtract $3$ from $0$ .

$\frac{x^{2}}{9} + \frac{{(- 3)}^{2}}{25} = 1$

Step 1.2.1.2

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

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

Step 1.2.2

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

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Step 1.2.2.1

Subtract $\frac{9}{25}$ from both sides of the equation.

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

Step 1.2.2.2

Write $1$ as a fraction with a common denominator.

$\frac{x^{2}}{9} = \frac{25}{25} - \frac{9}{25}$

Step 1.2.2.3

Combine the numerators over the common denominator.

$\frac{x^{2}}{9} = \frac{25 - 9}{25}$

Step 1.2.2.4

Subtract $9$ from $25$ .

$\frac{x^{2}}{9} = \frac{16}{25}$

Step 1.2.3

Multiply both sides of the equation by $9$ .

$9 \frac{x^{2}}{9} = 9 (\frac{16}{25})$

Step 1.2.4

Simplify both sides of the equation.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Cancel the common factor of $9$ .

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Step 1.2.4.1.1.1

Cancel the common factor.

$9 \frac{x^{2}}{9} = 9 (\frac{16}{25})$

Step 1.2.4.1.1.2

Rewrite the expression.

$x^{2} = 9 (\frac{16}{25})$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $9 (\frac{16}{25})$ .

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Step 1.2.4.2.1.1

Combine $9$ and $\frac{16}{25}$ .

$x^{2} = \frac{9 \cdot 16}{25}$

Step 1.2.4.2.1.2

Multiply $9$ by $16$ .

$x^{2} = \frac{144}{25}$

Step 1.2.5

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

$x = \pm \sqrt{\frac{144}{25}}$

Step 1.2.6

Simplify $\pm \sqrt{\frac{144}{25}}$ .

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Step 1.2.6.1

Rewrite $\sqrt{\frac{144}{25}}$ as $\frac{\sqrt{144}}{\sqrt{25}}$ .

$x = \pm \frac{\sqrt{144}}{\sqrt{25}}$

Step 1.2.6.2

Simplify the numerator.

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Step 1.2.6.2.1

Rewrite $144$ as $12^{2}$ .

$x = \pm \frac{\sqrt{12^{2}}}{\sqrt{25}}$

Step 1.2.6.2.2

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

$x = \pm \frac{12}{\sqrt{25}}$

Step 1.2.6.3

Simplify the denominator.

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Step 1.2.6.3.1

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

$x = \pm \frac{12}{\sqrt{5^{2}}}$

Step 1.2.6.3.2

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

$x = \pm \frac{12}{5}$

Step 1.2.7

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

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Step 1.2.7.1

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

$x = \frac{12}{5}$

Step 1.2.7.2

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

$x = - \frac{12}{5}$

Step 1.2.7.3

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

$x = \frac{12}{5}, - \frac{12}{5}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{12}{5}, 0), (- \frac{12}{5}, 0)$

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 $y$ .

$\frac{{(0)}^{2}}{9} + \frac{{(y - 3)}^{2}}{25} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Subtract $\frac{{(0)}^{2}}{9}$ from both sides of the equation.

$\frac{{(y - 3)}^{2}}{25} = 1 - \frac{{(0)}^{2}}{9}$

Step 2.2.2

Simplify $1 - \frac{{(0)}^{2}}{9}$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$\frac{{(y - 3)}^{2}}{25} = 1 - \frac{0}{9}$

Step 2.2.2.1.2

Divide $0$ by $9$ .

$\frac{{(y - 3)}^{2}}{25} = 1 - 0$

Step 2.2.2.1.3

Multiply $- 1$ by $0$ .

$\frac{{(y - 3)}^{2}}{25} = 1 + 0$

Step 2.2.2.2

Add $1$ and $0$ .

$\frac{{(y - 3)}^{2}}{25} = 1$

Step 2.2.3

Multiply both sides of the equation by $25$ .

$25 \frac{{(y - 3)}^{2}}{25} = 25 \cdot 1$

Step 2.2.4

Simplify both sides of the equation.

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Step 2.2.4.1

Simplify the left side.

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Step 2.2.4.1.1

Cancel the common factor of $25$ .

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Step 2.2.4.1.1.1

Cancel the common factor.

$25 \frac{{(y - 3)}^{2}}{25} = 25 \cdot 1$

Step 2.2.4.1.1.2

Rewrite the expression.

${(y - 3)}^{2} = 25 \cdot 1$

Step 2.2.4.2

Simplify the right side.

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Step 2.2.4.2.1

Multiply $25$ by $1$ .

${(y - 3)}^{2} = 25$

Step 2.2.5

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

$y - 3 = \pm \sqrt{25}$

Step 2.2.6

Simplify $\pm \sqrt{25}$ .

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Step 2.2.6.1

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

$y - 3 = \pm \sqrt{5^{2}}$

Step 2.2.6.2

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

$y - 3 = \pm 5$

Step 2.2.7

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

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Step 2.2.7.1

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

$y - 3 = 5$

Step 2.2.7.2

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

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Step 2.2.7.2.1

Add $3$ to both sides of the equation.

$y = 5 + 3$

Step 2.2.7.2.2

Add $5$ and $3$ .

$y = 8$

Step 2.2.7.3

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

$y - 3 = - 5$

Step 2.2.7.4

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

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Step 2.2.7.4.1

Add $3$ to both sides of the equation.

$y = - 5 + 3$

Step 2.2.7.4.2

Add $- 5$ and $3$ .

$y = - 2$

Step 2.2.7.5

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

$y = 8, - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8), (0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{12}{5}, 0), (- \frac{12}{5}, 0)$

y-intercept(s): $(0, 8), (0, - 2)$

Step 4