Calculus Examples

$\frac{{(x - 3)}^{2}}{100} + \frac{{(y - 4)}^{2}}{36} = 1$

Step 1

Subtract $\frac{{(x - 3)}^{2}}{100}$ from both sides of the equation.

$\frac{{(y - 4)}^{2}}{36} = 1 - \frac{{(x - 3)}^{2}}{100}$

Step 2

Simplify $1 - \frac{{(x - 3)}^{2}}{100}$ .

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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.

$\frac{{(y - 4)}^{2}}{36} = \frac{100}{100} - \frac{{(x - 3)}^{2}}{100}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(y - 4)}^{2}}{36} = \frac{100 - {(x - 3)}^{2}}{100}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $100$ as $10^{2}$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{10^{2} - {(x - 3)}^{2}}{100}$

Step 2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = x - 3$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(10 + x - 3) (10 - (x - 3))}{100}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Subtract $3$ from $10$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (10 - (x - 3))}{100}$

Step 2.2.3.2

Apply the distributive property.

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (10 - x - - 3)}{100}$

Step 2.2.3.3

Multiply $- 1$ by $- 3$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (10 - x + 3)}{100}$

Step 2.2.3.4

Add $10$ and $3$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (- x + 13)}{100}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Factor $- 1$ out of $- x$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (- (x) + 13)}{100}$

Step 2.3.2

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (- (x) - 1 (- 13))}{100}$

Step 2.3.3

Factor $- 1$ out of $- (x) - 1 (- 13)$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (- (x - 13))}{100}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Rewrite $- (x - 13)$ as $- 1 (x - 13)$ .

$\frac{{(y - 4)}^{2}}{36} = \frac{(x + 7) (- 1 (x - 13))}{100}$

Step 2.3.4.2

Move the negative in front of the fraction.

$\frac{{(y - 4)}^{2}}{36} = - \frac{(x + 7) (x - 13)}{100}$

Step 3

Multiply both sides of the equation by $36$ .

$36 \frac{{(y - 4)}^{2}}{36} = 36 (- \frac{(x + 7) (x - 13)}{100})$

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 $36$ .

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Step 4.1.1.1

Cancel the common factor.

$36 \frac{{(y - 4)}^{2}}{36} = 36 (- \frac{(x + 7) (x - 13)}{100})$

Step 4.1.1.2

Rewrite the expression.

${(y - 4)}^{2} = 36 (- \frac{(x + 7) (x - 13)}{100})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $36 (- \frac{(x + 7) (x - 13)}{100})$ .

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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

Move the leading negative in $- \frac{(x + 7) (x - 13)}{100}$ into the numerator.

${(y - 4)}^{2} = 36 \frac{- (x + 7) (x - 13)}{100}$

Step 4.2.1.1.2

Factor $4$ out of $36$ .

${(y - 4)}^{2} = 4 (9) \frac{- (x + 7) (x - 13)}{100}$

Step 4.2.1.1.3

Factor $4$ out of $100$ .

${(y - 4)}^{2} = 4 \cdot 9 \frac{- (x + 7) (x - 13)}{4 \cdot 25}$

Step 4.2.1.1.4

Cancel the common factor.

${(y - 4)}^{2} = 4 \cdot 9 \frac{- (x + 7) (x - 13)}{4 \cdot 25}$

Step 4.2.1.1.5

Rewrite the expression.

${(y - 4)}^{2} = 9 \frac{- (x + 7) (x - 13)}{25}$

Step 4.2.1.2

Combine $9$ and $\frac{- (x + 7) (x - 13)}{25}$ .

${(y - 4)}^{2} = \frac{9 (- (x + 7) (x - 13))}{25}$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Multiply $- 1$ by $9$ .

${(y - 4)}^{2} = \frac{- 9 ((x + 7) (x - 13))}{25}$

Step 4.2.1.3.2

Move the negative in front of the fraction.

${(y - 4)}^{2} = - \frac{9 (x + 7) (x - 13)}{25}$

Step 5

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

$y - 4 = \pm \sqrt{- \frac{9 (x + 7) (x - 13)}{25}}$

Step 6

Simplify $\pm \sqrt{- \frac{9 (x + 7) (x - 13)}{25}}$ .

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Step 6.1

Rewrite $- \frac{9 (x + 7) (x - 13)}{25}$ as ${(\frac{3}{5})}^{2} \cdot (- (x + 7) (x - 13))$ .

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Step 6.1.1

Factor the perfect power $3^{2}$ out of $9 (x + 7) (x - 13)$ .

$y - 4 = \pm \sqrt{- \frac{3^{2} ((x + 7) (x - 13))}{25}}$

Step 6.1.2

Factor the perfect power $5^{2}$ out of $25$ .

$y - 4 = \pm \sqrt{- \frac{3^{2} ((x + 7) (x - 13))}{5^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{3^{2} ((x + 7) (x - 13))}{5^{2} \cdot 1}$ .

$y - 4 = \pm \sqrt{- ({(\frac{3}{5})}^{2} ((x + 7) (x - 13)))}$

Step 6.1.4

Reorder $- 1$ and ${(\frac{3}{5})}^{2}$ .

$y - 4 = \pm \sqrt{{(\frac{3}{5})}^{2} \cdot - 1 (x + 7) (x - 13)}$

Step 6.1.5

Add parentheses.

$y - 4 = \pm \sqrt{{(\frac{3}{5})}^{2} \cdot - 1 ((x + 7) (x - 13))}$

Step 6.1.6

Add parentheses.

$y - 4 = \pm \sqrt{{(\frac{3}{5})}^{2} \cdot (- (x + 7) (x - 13))}$

Step 6.2

Pull terms out from under the radical.

$y - 4 = \pm \frac{3}{5} \sqrt{- (x + 7) (x - 13)}$

Step 6.3

Combine $\frac{3}{5}$ and $\sqrt{- (x + 7) (x - 13)}$ .

$y - 4 = \pm \frac{3 \sqrt{- (x + 7) (x - 13)}}{5}$

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.

$y - 4 = \frac{3 \sqrt{- (x + 7) (x - 13)}}{5}$

Step 7.2

Add $4$ to both sides of the equation.

$y = \frac{3 \sqrt{- (x + 7) (x - 13)}}{5} + 4$

Step 7.3

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

$y - 4 = - \frac{3 \sqrt{- (x + 7) (x - 13)}}{5}$

Step 7.4

Add $4$ to both sides of the equation.

$y = - \frac{3 \sqrt{- (x + 7) (x - 13)}}{5} + 4$

Step 7.5

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

$y = \frac{3 \sqrt{- (x + 7) (x - 13)}}{5} + 4$

$y = - \frac{3 \sqrt{- (x + 7) (x - 13)}}{5} + 4$

$y = \frac{3 \sqrt{- (x + 7) (x - 13)}}{5} + 4$

$y = - \frac{3 \sqrt{- (x + 7) (x - 13)}}{5} + 4$