Precalculus Examples

$x^{2} + y^{2} = 25$ , $3 x^{2} - 6 y = 0$

Step 1

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

$y^{2} = 25 - x^{2}$

$3 x^{2} - 6 y = 0$

Step 2

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

$y = \pm \sqrt{25 - x^{2}}$

$3 x^{2} - 6 y = 0$

Step 3

Simplify $\pm \sqrt{25 - x^{2}}$ .

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Rewrite $25$ as $5^{2}$ .

$y = \pm \sqrt{5^{2} - x^{2}}$

$3 x^{2} - 6 y = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$y = \pm \sqrt{(5 + x) (5 - x)}$

$3 x^{2} - 6 y = 0$

$y = \pm \sqrt{(5 + x) (5 - x)}$

$3 x^{2} - 6 y = 0$

Step 4

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

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

$y = \sqrt{(5 + x) (5 - x)}$

$3 x^{2} - 6 y = 0$

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

$y = - \sqrt{(5 + x) (5 - x)}$

$3 x^{2} - 6 y = 0$

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

$3 x^{2} - 6 y = 0$

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

$3 x^{2} - 6 y = 0$

Step 5

Subtract $3 x^{2}$ from both sides of the equation.

$- 6 y = - 3 x^{2}$

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Step 6

Divide each term in $- 6 y = - 3 x^{2}$ by $- 6$ and simplify.

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Divide each term in $- 6 y = - 3 x^{2}$ by $- 6$ .

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Simplify the left side.

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Cancel the common factor of $- 6$ .

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Cancel the common factor.

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Divide $y$ by $1$ .

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

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

$y = \sqrt{(5 + x) (5 - x)}, - \sqrt{(5 + x) (5 - x)}$

Simplify the right side.

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Cancel the common factor of $- 3$ and $- 6$ .

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Factor $- 3$ out of $- 3 x^{2}$ .

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Cancel the common factors.

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Factor $- 3$ out of $- 6$ .

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Cancel the common factor.

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

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Rewrite the expression.

$y = \frac{x^{2}}{2}$

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

$y = \frac{x^{2}}{2}$

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

$y = \frac{x^{2}}{2}$

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

$y = \frac{x^{2}}{2}$

$y = \sqrt{(5 + x) (5 - x)}, - \sqrt{(5 + x) (5 - x)}$

$y = \frac{x^{2}}{2}$

$y = \sqrt{(5 + x) (5 - x)}$

$y = - \sqrt{(5 + x) (5 - x)}$

Step 7

Create a graph to locate the intersection of the equations. The intersection of the system of equations is the solution.

$(- 2.86322179, 4.09901951)$

$(2.86322179, 4.09901951)$

Step 8