Calculus Examples

$\frac{{(x + 3)}^{2}}{64} - \frac{{(y + 1)}^{2}}{36} = 1$

Step 1

Add $\frac{{(y + 1)}^{2}}{36}$ to both sides of the equation.

$\frac{{(x + 3)}^{2}}{64} = 1 + \frac{{(y + 1)}^{2}}{36}$

Step 2

Simplify $1 + \frac{{(y + 1)}^{2}}{36}$ .

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Step 2.1

Combine into one fraction.

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Step 2.1.1

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

$\frac{{(x + 3)}^{2}}{64} = \frac{36}{36} + \frac{{(y + 1)}^{2}}{36}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + {(y + 1)}^{2}}{36}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite ${(y + 1)}^{2}$ as $(y + 1) (y + 1)$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + (y + 1) (y + 1)}{36}$

Step 2.2.2

Expand $(y + 1) (y + 1)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y (y + 1) + 1 (y + 1)}{36}$

Step 2.2.2.2

Apply the distributive property.

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y \cdot y + y \cdot 1 + 1 (y + 1)}{36}$

Step 2.2.2.3

Apply the distributive property.

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1}{36}$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $y$ by $y$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y^{2} + y \cdot 1 + 1 y + 1 \cdot 1}{36}$

Step 2.2.3.1.2

Multiply $y$ by $1$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y^{2} + y + 1 y + 1 \cdot 1}{36}$

Step 2.2.3.1.3

Multiply $y$ by $1$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y^{2} + y + y + 1 \cdot 1}{36}$

Step 2.2.3.1.4

Multiply $1$ by $1$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y^{2} + y + y + 1}{36}$

Step 2.2.3.2

Add $y$ and $y$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{36 + y^{2} + 2 y + 1}{36}$

Step 2.2.4

Add $36$ and $1$ .

$\frac{{(x + 3)}^{2}}{64} = \frac{y^{2} + 2 y + 37}{36}$

Step 3

Multiply both sides of the equation by $64$ .

$64 \frac{{(x + 3)}^{2}}{64} = 64 \frac{y^{2} + 2 y + 37}{36}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $64$ .

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Step 4.1.1.1

Cancel the common factor.

$64 \frac{{(x + 3)}^{2}}{64} = 64 \frac{y^{2} + 2 y + 37}{36}$

Step 4.1.1.2

Rewrite the expression.

${(x + 3)}^{2} = 64 \frac{y^{2} + 2 y + 37}{36}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $64 \frac{y^{2} + 2 y + 37}{36}$ .

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Factor $4$ out of $64$ .

${(x + 3)}^{2} = 4 (16) \frac{y^{2} + 2 y + 37}{36}$

Step 4.2.1.1.2

Factor $4$ out of $36$ .

${(x + 3)}^{2} = 4 \cdot 16 \frac{y^{2} + 2 y + 37}{4 \cdot 9}$

Step 4.2.1.1.3

Cancel the common factor.

${(x + 3)}^{2} = 4 \cdot 16 \frac{y^{2} + 2 y + 37}{4 \cdot 9}$

Step 4.2.1.1.4

Rewrite the expression.

${(x + 3)}^{2} = 16 \frac{y^{2} + 2 y + 37}{9}$

Step 4.2.1.2

Combine $16$ and $\frac{y^{2} + 2 y + 37}{9}$ .

${(x + 3)}^{2} = \frac{16 (y^{2} + 2 y + 37)}{9}$

Step 5

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

$x + 3 = \pm \sqrt{\frac{16 (y^{2} + 2 y + 37)}{9}}$

Step 6

Simplify $\pm \sqrt{\frac{16 (y^{2} + 2 y + 37)}{9}}$ .

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Step 6.1

Rewrite $\frac{16 (y^{2} + 2 y + 37)}{9}$ as ${(\frac{2^{2}}{3})}^{2} (y^{2} + 2 y + 37)$ .

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Step 6.1.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (y^{2} + 2 y + 37)$ .

$x + 3 = \pm \sqrt{\frac{{(2^{2})}^{2} (y^{2} + 2 y + 37)}{9}}$

Step 6.1.2

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

$x + 3 = \pm \sqrt{\frac{{(2^{2})}^{2} (y^{2} + 2 y + 37)}{3^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{{(2^{2})}^{2} (y^{2} + 2 y + 37)}{3^{2} \cdot 1}$ .

$x + 3 = \pm \sqrt{{(\frac{2^{2}}{3})}^{2} (y^{2} + 2 y + 37)}$

Step 6.2

Pull terms out from under the radical.

$x + 3 = \pm \frac{2^{2}}{3} \sqrt{y^{2} + 2 y + 37}$

Step 6.3

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

$x + 3 = \pm \frac{4}{3} \sqrt{y^{2} + 2 y + 37}$

Step 6.4

Combine $\frac{4}{3}$ and $\sqrt{y^{2} + 2 y + 37}$ .

$x + 3 = \pm \frac{4 \sqrt{y^{2} + 2 y + 37}}{3}$

Step 7

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

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Step 7.1

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

$x + 3 = \frac{4 \sqrt{y^{2} + 2 y + 37}}{3}$

Step 7.2

Subtract $3$ from both sides of the equation.

$x = \frac{4 \sqrt{y^{2} + 2 y + 37}}{3} - 3$

Step 7.3

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

$x + 3 = - \frac{4 \sqrt{y^{2} + 2 y + 37}}{3}$

Step 7.4

Subtract $3$ from both sides of the equation.

$x = - \frac{4 \sqrt{y^{2} + 2 y + 37}}{3} - 3$

Step 7.5

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

$x = \frac{4 \sqrt{y^{2} + 2 y + 37}}{3} - 3$

$x = - \frac{4 \sqrt{y^{2} + 2 y + 37}}{3} - 3$

$x = \frac{4 \sqrt{y^{2} + 2 y + 37}}{3} - 3$

$x = - \frac{4 \sqrt{y^{2} + 2 y + 37}}{3} - 3$