Precalculus Examples

$2 x^{2} + 3 y^{2} = 1$

Step 1

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

$3 y^{2} = 1 - 2 x^{2}$

Step 2

Divide each term in $3 y^{2} = 1 - 2 x^{2}$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 y^{2} = 1 - 2 x^{2}$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{1}{3} + \frac{- 2 x^{2}}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{1}{3} + \frac{- 2 x^{2}}{3}$

Step 2.2.1.2

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

$y^{2} = \frac{1}{3} + \frac{- 2 x^{2}}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y^{2} = \frac{1}{3} - \frac{2 x^{2}}{3}$

Step 3

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

$y = \pm \sqrt{\frac{1}{3} - \frac{2 x^{2}}{3}}$

Step 4

Simplify $\pm \sqrt{\frac{1}{3} - \frac{2 x^{2}}{3}}$ .

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Step 4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{1 - 2 x^{2}}{3}}$

Step 4.2

Rewrite $\sqrt{\frac{1 - 2 x^{2}}{3}}$ as $\frac{\sqrt{1 - 2 x^{2}}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{1 - 2 x^{2}}}{\sqrt{3}}$

Step 4.3

Multiply $\frac{\sqrt{1 - 2 x^{2}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{\sqrt{1 - 2 x^{2}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{3^{1}}$

Step 4.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{1 - 2 x^{2}} \sqrt{3}}{3}$

Step 4.5

Combine using the product rule for radicals.

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

Step 4.6

Reorder factors in $\pm \frac{\sqrt{(1 - 2 x^{2}) \cdot 3}}{3}$ .

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

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.

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

Step 5.2

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

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

Step 5.3

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

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

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

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

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

Step 6

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{\sqrt{3 (1 - 2 x^{2})}}{3}, - \frac{\sqrt{3 (1 - 2 x^{2})}}{3}$