Algebra Examples

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

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 $x$ .

$\frac{{(x - 2)}^{2}}{9} - \frac{{((0) - 1)}^{2}}{4} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Add $\frac{{((0) - 1)}^{2}}{4}$ to both sides of the equation.

$\frac{{(x - 2)}^{2}}{9} = 1 + \frac{{((0) - 1)}^{2}}{4}$

Step 1.2.2

Simplify $1 + \frac{{((0) - 1)}^{2}}{4}$ .

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Step 1.2.2.1

Subtract $1$ from $0$ .

$\frac{{(x - 2)}^{2}}{9} = 1 + \frac{{(- 1)}^{2}}{4}$

Step 1.2.2.2

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

$\frac{{(x - 2)}^{2}}{9} = 1 + \frac{1}{4}$

Step 1.2.2.3

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

$\frac{{(x - 2)}^{2}}{9} = \frac{4}{4} + \frac{1}{4}$

Step 1.2.2.4

Combine the numerators over the common denominator.

$\frac{{(x - 2)}^{2}}{9} = \frac{4 + 1}{4}$

Step 1.2.2.5

Add $4$ and $1$ .

$\frac{{(x - 2)}^{2}}{9} = \frac{5}{4}$

Step 1.2.3

Multiply both sides of the equation by $9$ .

$9 \frac{{(x - 2)}^{2}}{9} = 9 (\frac{5}{4})$

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.

$9 \frac{{(x - 2)}^{2}}{9} = 9 (\frac{5}{4})$

Step 1.2.4.1.1.2

Rewrite the expression.

${(x - 2)}^{2} = 9 (\frac{5}{4})$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $9 (\frac{5}{4})$ .

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Step 1.2.4.2.1.1

Combine $9$ and $\frac{5}{4}$ .

${(x - 2)}^{2} = \frac{9 \cdot 5}{4}$

Step 1.2.4.2.1.2

Multiply $9$ by $5$ .

${(x - 2)}^{2} = \frac{45}{4}$

Step 1.2.5

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

$x - 2 = \pm \sqrt{\frac{45}{4}}$

Step 1.2.6

Simplify $\pm \sqrt{\frac{45}{4}}$ .

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Step 1.2.6.1

Rewrite $\sqrt{\frac{45}{4}}$ as $\frac{\sqrt{45}}{\sqrt{4}}$ .

$x - 2 = \pm \frac{\sqrt{45}}{\sqrt{4}}$

Step 1.2.6.2

Simplify the numerator.

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Step 1.2.6.2.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.2.6.2.1.1

Factor $9$ out of $45$ .

$x - 2 = \pm \frac{\sqrt{9 (5)}}{\sqrt{4}}$

Step 1.2.6.2.1.2

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

$x - 2 = \pm \frac{\sqrt{3^{2} \cdot 5}}{\sqrt{4}}$

Step 1.2.6.2.2

Pull terms out from under the radical.

$x - 2 = \pm \frac{3 \sqrt{5}}{\sqrt{4}}$

Step 1.2.6.3

Simplify the denominator.

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Step 1.2.6.3.1

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

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

Step 1.2.6.3.2

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

$x - 2 = \pm \frac{3 \sqrt{5}}{2}$

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.

$x - 2 = \frac{3 \sqrt{5}}{2}$

Step 1.2.7.2

Add $2$ to both sides of the equation.

$x = \frac{3 \sqrt{5}}{2} + 2$

Step 1.2.7.3

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

$x - 2 = - \frac{3 \sqrt{5}}{2}$

Step 1.2.7.4

Add $2$ to both sides of the equation.

$x = - \frac{3 \sqrt{5}}{2} + 2$

Step 1.2.7.5

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

$x = \frac{3 \sqrt{5}}{2} + 2, - \frac{3 \sqrt{5}}{2} + 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3 \sqrt{5}}{2} + 2, 0), (- \frac{3 \sqrt{5}}{2} + 2, 0)$

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 $y$ .

$\frac{{((0) - 2)}^{2}}{9} - \frac{{(y - 1)}^{2}}{4} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

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

$- \frac{{(y - 1)}^{2}}{4} = 1 - \frac{{((0) - 2)}^{2}}{9}$

Step 2.2.2

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

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Step 2.2.2.1

Simplify the numerator.

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Step 2.2.2.1.1

Subtract $2$ from $0$ .

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

Step 2.2.2.1.2

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

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

Step 2.2.2.2

Simplify the expression.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.2.2.3

Subtract $4$ from $9$ .

$- \frac{{(y - 1)}^{2}}{4} = \frac{5}{9}$

Step 2.2.3

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{{(y - 1)}^{2}}{4}) = - 4 (\frac{5}{9})$

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

Simplify $- 4 (- \frac{{(y - 1)}^{2}}{4})$ .

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Step 2.2.4.1.1.1

Cancel the common factor of $4$ .

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Step 2.2.4.1.1.1.1

Move the leading negative in $- \frac{{(y - 1)}^{2}}{4}$ into the numerator.

$- 4 \frac{- {(y - 1)}^{2}}{4} = - 4 (\frac{5}{9})$

Step 2.2.4.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- {(y - 1)}^{2}}{4} = - 4 (\frac{5}{9})$

Step 2.2.4.1.1.1.3

Cancel the common factor.

$4 \cdot - 1 \frac{- {(y - 1)}^{2}}{4} = - 4 (\frac{5}{9})$

Step 2.2.4.1.1.1.4

Rewrite the expression.

$- - {(y - 1)}^{2} = - 4 (\frac{5}{9})$

Step 2.2.4.1.1.2

Multiply.

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Step 2.2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 {(y - 1)}^{2} = - 4 (\frac{5}{9})$

Step 2.2.4.1.1.2.2

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

${(y - 1)}^{2} = - 4 (\frac{5}{9})$

Step 2.2.4.2

Simplify the right side.

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Step 2.2.4.2.1

Simplify $- 4 (\frac{5}{9})$ .

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Step 2.2.4.2.1.1

Multiply $- 4 (\frac{5}{9})$ .

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Step 2.2.4.2.1.1.1

Combine $- 4$ and $\frac{5}{9}$ .

${(y - 1)}^{2} = \frac{- 4 \cdot 5}{9}$

Step 2.2.4.2.1.1.2

Multiply $- 4$ by $5$ .

${(y - 1)}^{2} = \frac{- 20}{9}$

Step 2.2.4.2.1.2

Move the negative in front of the fraction.

${(y - 1)}^{2} = - \frac{20}{9}$

Step 2.2.5

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

$y - 1 = \pm \sqrt{- \frac{20}{9}}$

Step 2.2.6

Simplify $\pm \sqrt{- \frac{20}{9}}$ .

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Step 2.2.6.1

Rewrite $- \frac{20}{9}$ as ${(\frac{2 i}{3})}^{2} \cdot 5$ .

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Step 2.2.6.1.1

Rewrite $- 1$ as $i^{2}$ .

$y - 1 = \pm \sqrt{i^{2} \frac{20}{9}}$

Step 2.2.6.1.2

Factor the perfect power $2^{2}$ out of $20$ .

$y - 1 = \pm \sqrt{i^{2} \frac{2^{2} \cdot 5}{9}}$

Step 2.2.6.1.3

Factor the perfect power $3^{2}$ out of $9$ .

$y - 1 = \pm \sqrt{i^{2} \frac{2^{2} \cdot 5}{3^{2} \cdot 1}}$

Step 2.2.6.1.4

Rearrange the fraction $\frac{2^{2} \cdot 5}{3^{2} \cdot 1}$ .

$y - 1 = \pm \sqrt{i^{2} ({(\frac{2}{3})}^{2} \cdot 5)}$

Step 2.2.6.1.5

Rewrite $i^{2} {(\frac{2}{3})}^{2}$ as ${(\frac{2 i}{3})}^{2}$ .

$y - 1 = \pm \sqrt{{(\frac{2 i}{3})}^{2} \cdot 5}$

Step 2.2.6.2

Pull terms out from under the radical.

$y - 1 = \pm \frac{2 i}{3} \sqrt{5}$

Step 2.2.6.3

Combine $\frac{2 i}{3}$ and $\sqrt{5}$ .

$y - 1 = \pm \frac{2 i \sqrt{5}}{3}$

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.

$y - 1 = \frac{2 i \sqrt{5}}{3}$

Step 2.2.7.2

Add $1$ to both sides of the equation.

$y = \frac{2 i \sqrt{5}}{3} + 1$

Step 2.2.7.3

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

$y - 1 = - \frac{2 i \sqrt{5}}{3}$

Step 2.2.7.4

Add $1$ to both sides of the equation.

$y = - \frac{2 i \sqrt{5}}{3} + 1$

Step 2.2.7.5

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

$y = \frac{2 i \sqrt{5}}{3} + 1, - \frac{2 i \sqrt{5}}{3} + 1$

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): $(\frac{3 \sqrt{5}}{2} + 2, 0), (- \frac{3 \sqrt{5}}{2} + 2, 0)$

y-intercept(s): $None$

Step 4