Calculus Examples

$\frac{x^{2}}{9} = \frac{x^{2}}{36} + h$

Step 1

Multiply both sides by $9$ .

$\frac{x^{2}}{9} \cdot 9 = (\frac{x^{2}}{36} + h) \cdot 9$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $9$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x^{2}}{9} \cdot 9 = (\frac{x^{2}}{36} + h) \cdot 9$

Step 2.1.1.2

Rewrite the expression.

$x^{2} = (\frac{x^{2}}{36} + h) \cdot 9$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{x^{2}}{36} + h) \cdot 9$ .

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Step 2.2.1.1

Apply the distributive property.

$x^{2} = \frac{x^{2}}{36} \cdot 9 + h \cdot 9$

Step 2.2.1.2

Cancel the common factor of $9$ .

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Step 2.2.1.2.1

Factor $9$ out of $36$ .

$x^{2} = \frac{x^{2}}{9 (4)} \cdot 9 + h \cdot 9$

Step 2.2.1.2.2

Cancel the common factor.

$x^{2} = \frac{x^{2}}{9 \cdot 4} \cdot 9 + h \cdot 9$

Step 2.2.1.2.3

Rewrite the expression.

$x^{2} = \frac{x^{2}}{4} + h \cdot 9$

Step 2.2.1.3

Move $9$ to the left of $h$ .

$x^{2} = \frac{x^{2}}{4} + 9 h$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $\frac{x^{2}}{4}$ from both sides of the equation.

$x^{2} - \frac{x^{2}}{4} = 9 h$

Step 3.1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} \cdot \frac{4}{4} - \frac{x^{2}}{4} = 9 h$

Step 3.1.3

Combine $x^{2}$ and $\frac{4}{4}$ .

$\frac{x^{2} \cdot 4}{4} - \frac{x^{2}}{4} = 9 h$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 - x^{2}}{4} = 9 h$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Move $4$ to the left of $x^{2}$ .

$\frac{4 \cdot x^{2} - x^{2}}{4} = 9 h$

Step 3.1.5.2

Subtract $x^{2}$ from $4 x^{2}$ .

$\frac{3 x^{2}}{4} = 9 h$

Step 3.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x^{2}}{4} = \frac{4}{3} (9 h)$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x^{2}}{4}$ .

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Step 3.3.1.1.1

Combine.

$\frac{4 (3 x^{2})}{3 \cdot 4} = \frac{4}{3} (9 h)$

Step 3.3.1.1.2

Cancel the common factor of $4$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{4 (3 x^{2})}{3 \cdot 4} = \frac{4}{3} (9 h)$

Step 3.3.1.1.2.2

Rewrite the expression.

$\frac{3 x^{2}}{3} = \frac{4}{3} (9 h)$

Step 3.3.1.1.3

Cancel the common factor of $3$ .

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Step 3.3.1.1.3.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{4}{3} (9 h)$

Step 3.3.1.1.3.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{4}{3} (9 h)$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{4}{3} (9 h)$ .

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Step 3.3.2.1.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.1

Factor $3$ out of $9 h$ .

$x^{2} = \frac{4}{3} (3 (3 h))$

Step 3.3.2.1.1.2

Cancel the common factor.

$x^{2} = \frac{4}{3} (3 (3 h))$

Step 3.3.2.1.1.3

Rewrite the expression.

$x^{2} = 4 (3 h)$

Step 3.3.2.1.2

Multiply $3$ by $4$ .

$x^{2} = 12 h$

Step 3.4

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

$x = \pm \sqrt{12 h}$

Step 3.5

Simplify $\pm \sqrt{12 h}$ .

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Step 3.5.1

Rewrite $12 h$ as $2^{2} \cdot (3 h)$ .

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Step 3.5.1.1

Factor $4$ out of $12$ .

$x = \pm \sqrt{4 (3) h}$

Step 3.5.1.2

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

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

Step 3.5.1.3

Add parentheses.

$x = \pm \sqrt{2^{2} \cdot (3 h)}$

Step 3.5.2

Pull terms out from under the radical.

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

Step 3.6

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

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Step 3.6.1

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

$x = 2 \sqrt{3 h}$

Step 3.6.2

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

$x = - 2 \sqrt{3 h}$

Step 3.6.3

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

$x = 2 \sqrt{3 h}$

$x = - 2 \sqrt{3 h}$

$x = 2 \sqrt{3 h}$

$x = - 2 \sqrt{3 h}$

$x = 2 \sqrt{3 h}$

$x = - 2 \sqrt{3 h}$