Calculus Examples

$9 x^{2} + 164 y^{2} = 144$

Step 1

Subtract $164 y^{2}$ from both sides of the equation.

$9 x^{2} = 144 - 164 y^{2}$

Step 2

Divide each term in $9 x^{2} = 144 - 164 y^{2}$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 x^{2} = 144 - 164 y^{2}$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{144}{9} + \frac{- 164 y^{2}}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{144}{9} + \frac{- 164 y^{2}}{9}$

Step 2.2.1.2

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

$x^{2} = \frac{144}{9} + \frac{- 164 y^{2}}{9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $144$ by $9$ .

$x^{2} = 16 + \frac{- 164 y^{2}}{9}$

Step 2.3.1.2

Move the negative in front of the fraction.

$x^{2} = 16 - \frac{164 y^{2}}{9}$

Step 3

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

$x = \pm \sqrt{16 - \frac{164 y^{2}}{9}}$

Step 4

Simplify $\pm \sqrt{16 - \frac{164 y^{2}}{9}}$ .

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Step 4.1

Factor $4$ out of $16 - \frac{164 y^{2}}{9}$ .

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Step 4.1.1

Factor $4$ out of $16$ .

$x = \pm \sqrt{4 (4) - \frac{164 y^{2}}{9}}$

Step 4.1.2

Factor $4$ out of $- \frac{164 y^{2}}{9}$ .

$x = \pm \sqrt{4 (4) + 4 (- \frac{41 y^{2}}{9})}$

Step 4.1.3

Factor $4$ out of $4 (4) + 4 (- \frac{41 y^{2}}{9})$ .

$x = \pm \sqrt{4 (4 - \frac{41 y^{2}}{9})}$

Step 4.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = \pm \sqrt{4 (4 \cdot \frac{9}{9} - \frac{41 y^{2}}{9})}$

Step 4.3

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

$x = \pm \sqrt{4 (\frac{4 \cdot 9}{9} - \frac{41 y^{2}}{9})}$

Step 4.4

Combine the numerators over the common denominator.

$x = \pm \sqrt{4 \frac{4 \cdot 9 - 41 y^{2}}{9}}$

Step 4.5

Multiply $4$ by $9$ .

$x = \pm \sqrt{4 \frac{36 - 41 y^{2}}{9}}$

Step 4.6

Combine $4$ and $\frac{36 - 41 y^{2}}{9}$ .

$x = \pm \sqrt{\frac{4 (36 - 41 y^{2})}{9}}$

Step 4.7

Rewrite $\frac{4 (36 - 41 y^{2})}{9}$ as ${(\frac{2}{3})}^{2} (6^{2} - 41 y^{2})$ .

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Step 4.7.1

Factor the perfect power $2^{2}$ out of $4 (36 - 41 y^{2})$ .

$x = \pm \sqrt{\frac{2^{2} (6^{2} - 41 y^{2})}{9}}$

Step 4.7.2

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

$x = \pm \sqrt{\frac{2^{2} (6^{2} - 41 y^{2})}{3^{2} \cdot 1}}$

Step 4.7.3

Rearrange the fraction $\frac{2^{2} (6^{2} - 41 y^{2})}{3^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{2}{3})}^{2} (6^{2} - 41 y^{2})}$

Step 4.8

Pull terms out from under the radical.

$x = \pm \frac{2}{3} \sqrt{6^{2} - 41 y^{2}}$

Step 4.9

Raise $6$ to the power of $2$ .

$x = \pm \frac{2}{3} \sqrt{36 - 41 y^{2}}$

Step 4.10

Combine $\frac{2}{3}$ and $\sqrt{36 - 41 y^{2}}$ .

$x = \pm \frac{2 \sqrt{36 - 41 y^{2}}}{3}$

Step 5

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

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Step 5.1

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

$x = \frac{2 \sqrt{36 - 41 y^{2}}}{3}$

Step 5.2

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

$x = - \frac{2 \sqrt{36 - 41 y^{2}}}{3}$

Step 5.3

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

$x = \frac{2 \sqrt{36 - 41 y^{2}}}{3}$

$x = - \frac{2 \sqrt{36 - 41 y^{2}}}{3}$

$x = \frac{2 \sqrt{36 - 41 y^{2}}}{3}$

$x = - \frac{2 \sqrt{36 - 41 y^{2}}}{3}$