Precalculus Examples

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

Step 1

Rewrite the equation as $3 - x^{2} = y^{2}$ .

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

Step 2

Subtract $3$ from both sides of the equation.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative one from the denominator of $\frac{y^{2}}{- 1}$ .

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

Step 3.3.1.2

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

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

Step 3.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 4

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

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

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.

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

Step 5.2

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

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

Step 5.3

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

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

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

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

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

Step 6

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

$f (y) = \sqrt{- y^{2} + 3}, - \sqrt{- y^{2} + 3}$