Precalculus Examples

$y = 4 - x^{2}$ , $y = x^{2} - 4$

Step 1

Eliminate the equal sides of each equation and combine.

$4 - x^{2} = x^{2} - 4$

Step 2

Solve $4 - x^{2} = x^{2} - 4$ for $x$ .

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Step 2.1

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

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Step 2.1.1

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

$4 - x^{2} - x^{2} = - 4$

Step 2.1.2

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

$4 - 2 x^{2} = - 4$

Step 2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.1

Subtract $4$ from both sides of the equation.

$- 2 x^{2} = - 4 - 4$

Step 2.2.2

Subtract $4$ from $- 4$ .

$- 2 x^{2} = - 8$

Step 2.3

Divide each term in $- 2 x^{2} = - 8$ by $- 2$ and simplify.

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Step 2.3.1

Divide each term in $- 2 x^{2} = - 8$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 8}{- 2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 8}{- 2}$

Step 2.3.2.1.2

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

$x^{2} = \frac{- 8}{- 2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 8$ by $- 2$ .

$x^{2} = 4$

Step 2.4

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

$x = \pm \sqrt{4}$

Step 2.5

Simplify $\pm \sqrt{4}$ .

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Step 2.5.1

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

$x = \pm \sqrt{2^{2}}$

Step 2.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 2.6

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

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Step 2.6.1

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

$x = 2$

Step 2.6.2

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

$x = - 2$

Step 2.6.3

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

$x = 2, - 2$

Step 3

Evaluate $y$ when $x = 2$ .

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Step 3.1

Substitute $2$ for $x$ .

$y = {(2)}^{2} - 4$

Step 3.2

Substitute $2$ for $x$ in $y = {(2)}^{2} - 4$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 2^{2} - 4$

Step 3.2.2

Simplify $2^{2} - 4$ .

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Step 3.2.2.1

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

$y = 4 - 4$

Step 3.2.2.2

Subtract $4$ from $4$ .

$y = 0$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, 0)$

$(- 2, 0)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(2, 0), (- 2, 0)$

Equation Form:

$x = 2, y = 0$

$x = - 2, y = 0$

Step 6