Precalculus Examples

$x^{2} + y^{2} = 16$ , $y^{2} - 2 x = 16$

Step 1

Solve for $x$ in $y^{2} - 2 x = 16$ .

Tap for more steps...

Step 1.1

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

$- 2 x = 16 - y^{2}$

$x^{2} + y^{2} = 16$

Step 1.2

Divide each term in $- 2 x = 16 - y^{2}$ by $- 2$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in $- 2 x = 16 - y^{2}$ by $- 2$ .

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

$x^{2} + y^{2} = 16$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

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

$x^{2} + y^{2} = 16$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$x^{2} + y^{2} = 16$

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

$x^{2} + y^{2} = 16$

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

$x^{2} + y^{2} = 16$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Divide $16$ by $- 2$ .

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

$x^{2} + y^{2} = 16$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

$x^{2} + y^{2} = 16$

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

$x^{2} + y^{2} = 16$

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

$x^{2} + y^{2} = 16$

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

$x^{2} + y^{2} = 16$

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

$x^{2} + y^{2} = 16$

Step 2

Replace all occurrences of $x$ with $- 8 + \frac{y^{2}}{2}$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 16$ with $- 8 + \frac{y^{2}}{2}$ .

${(- 8 + \frac{y^{2}}{2})}^{2} + y^{2} = 16$

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${(- 8 + \frac{y^{2}}{2})}^{2} + y^{2}$ .

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Rewrite ${(- 8 + \frac{y^{2}}{2})}^{2}$ as $(- 8 + \frac{y^{2}}{2}) (- 8 + \frac{y^{2}}{2})$ .

$(- 8 + \frac{y^{2}}{2}) (- 8 + \frac{y^{2}}{2}) + y^{2} = 16$

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

Step 2.2.1.1.2

Expand $(- 8 + \frac{y^{2}}{2}) (- 8 + \frac{y^{2}}{2})$ using the FOIL Method.

Tap for more steps...

Step 2.2.1.1.2.1

Apply the distributive property.

$- 8 (- 8 + \frac{y^{2}}{2}) + \frac{y^{2}}{2} \cdot (- 8 + \frac{y^{2}}{2}) + y^{2} = 16$

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

Step 2.2.1.1.2.2

Apply the distributive property.

$- 8 \cdot - 8 - 8 \frac{y^{2}}{2} + \frac{y^{2}}{2} \cdot (- 8 + \frac{y^{2}}{2}) + y^{2} = 16$

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

Step 2.2.1.1.2.3

Apply the distributive property.

$- 8 \cdot - 8 - 8 \frac{y^{2}}{2} + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

$- 8 \cdot - 8 - 8 \frac{y^{2}}{2} + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.2.1.1.3.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.3.1.1

Multiply $- 8$ by $- 8$ .

$64 - 8 \frac{y^{2}}{2} + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.3.1.2.1

Factor $2$ out of $- 8$ .

$64 + 2 (- 4) (\frac{y^{2}}{2}) + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.2.2

Cancel the common factor.

$64 + 2 \cdot (- 4 \frac{y^{2}}{2}) + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.2.3

Rewrite the expression.

$64 - 4 y^{2} + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

$64 - 4 y^{2} + \frac{y^{2}}{2} \cdot - 8 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.3.1.3.1

Factor $2$ out of $- 8$ .

$64 - 4 y^{2} + \frac{y^{2}}{2} \cdot (2 (- 4)) + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.3.2

Cancel the common factor.

$64 - 4 y^{2} + \frac{y^{2}}{2} \cdot (2 \cdot - 4) + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.3.3

Rewrite the expression.

$64 - 4 y^{2} + y^{2} \cdot - 4 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

$64 - 4 y^{2} + y^{2} \cdot - 4 + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.4

Move $- 4$ to the left of $y^{2}$ .

$64 - 4 y^{2} - 4 \cdot y^{2} + \frac{y^{2}}{2} \cdot \frac{y^{2}}{2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.5

Combine.

$64 - 4 y^{2} - 4 y^{2} + \frac{y^{2} y^{2}}{2 \cdot 2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.6

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Tap for more steps...

Step 2.2.1.1.3.1.6.1

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

$64 - 4 y^{2} - 4 y^{2} + \frac{y^{2 + 2}}{2 \cdot 2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.6.2

Add $2$ and $2$ .

$64 - 4 y^{2} - 4 y^{2} + \frac{y^{4}}{2 \cdot 2} + y^{2} = 16$

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

$64 - 4 y^{2} - 4 y^{2} + \frac{y^{4}}{2 \cdot 2} + y^{2} = 16$

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

Step 2.2.1.1.3.1.7

Multiply $2$ by $2$ .

$64 - 4 y^{2} - 4 y^{2} + \frac{y^{4}}{4} + y^{2} = 16$

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

$64 - 4 y^{2} - 4 y^{2} + \frac{y^{4}}{4} + y^{2} = 16$

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

Step 2.2.1.1.3.2

Subtract $4 y^{2}$ from $- 4 y^{2}$ .

$64 - 8 y^{2} + \frac{y^{4}}{4} + y^{2} = 16$

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

$64 - 8 y^{2} + \frac{y^{4}}{4} + y^{2} = 16$

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

$64 - 8 y^{2} + \frac{y^{4}}{4} + y^{2} = 16$

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

Step 2.2.1.2

Add $- 8 y^{2}$ and $y^{2}$ .

$64 + \frac{y^{4}}{4} - 7 y^{2} = 16$

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

$64 + \frac{y^{4}}{4} - 7 y^{2} = 16$

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

$64 + \frac{y^{4}}{4} - 7 y^{2} = 16$

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

$64 + \frac{y^{4}}{4} - 7 y^{2} = 16$

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

Step 3

Solve for $y$ in $64 + \frac{y^{4}}{4} - 7 y^{2} = 16$ .

Tap for more steps...

Step 3.1

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$\frac{u^{2}}{4} - 7 u + 64 = 16$

$u = y^{2}$

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

Step 3.2

Multiply each term in $\frac{u^{2}}{4} - 7 u + 64 = 16$ by $4$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $\frac{u^{2}}{4} - 7 u + 64 = 16$ by $4$ .

$\frac{u^{2}}{4} \cdot 4 - 7 u \cdot 4 + 64 \cdot 4 = 16 \cdot 4$

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{u^{2}}{4} \cdot 4 - 7 u \cdot 4 + 64 \cdot 4 = 16 \cdot 4$

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

Step 3.2.2.1.1.2

Rewrite the expression.

$u^{2} - 7 u \cdot 4 + 64 \cdot 4 = 16 \cdot 4$

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

$u^{2} - 7 u \cdot 4 + 64 \cdot 4 = 16 \cdot 4$

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

Step 3.2.2.1.2

Multiply $4$ by $- 7$ .

$u^{2} - 28 u + 64 \cdot 4 = 16 \cdot 4$

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

Step 3.2.2.1.3

Multiply $64$ by $4$ .

$u^{2} - 28 u + 256 = 16 \cdot 4$

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

$u^{2} - 28 u + 256 = 16 \cdot 4$

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

$u^{2} - 28 u + 256 = 16 \cdot 4$

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

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Multiply $16$ by $4$ .

$u^{2} - 28 u + 256 = 64$

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

$u^{2} - 28 u + 256 = 64$

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

$u^{2} - 28 u + 256 = 64$

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

Step 3.3

Subtract $64$ from both sides of the equation.

$u^{2} - 28 u + 256 - 64 = 0$

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

Step 3.4

Subtract $64$ from $256$ .

$u^{2} - 28 u + 192 = 0$

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

Step 3.5

Factor $u^{2} - 28 u + 192$ using the AC method.

Tap for more steps...

Step 3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $192$ and whose sum is $- 28$ .

$- 16, - 12$

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

Step 3.5.2

Write the factored form using these integers.

$(u - 16) (u - 12) = 0$

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

$(u - 16) (u - 12) = 0$

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

Step 3.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 16 = 0$

$u - 12 = 0$

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

Step 3.7

Set $u - 16$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 3.7.1

Set $u - 16$ equal to $0$ .

$u - 16 = 0$

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

Step 3.7.2

Add $16$ to both sides of the equation.

$u = 16$

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

$u = 16$

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

Step 3.8

Set $u - 12$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 3.8.1

Set $u - 12$ equal to $0$ .

$u - 12 = 0$

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

Step 3.8.2

Add $12$ to both sides of the equation.

$u = 12$

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

$u = 12$

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

Step 3.9

The final solution is all the values that make $(u - 16) (u - 12) = 0$ true.

$u = 16, 12$

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

Step 3.10

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 16$

$y^{2} = 12$

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

Step 3.11

Solve the first equation for $y$ .

$y^{2} = 16$

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

Step 3.12

Solve the equation for $y$ .

Tap for more steps...

Step 3.12.1

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

$y = \pm \sqrt{16}$

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

Step 3.12.2

Simplify $\pm \sqrt{16}$ .

Tap for more steps...

Step 3.12.2.1

Rewrite $16$ as $4^{2}$ .

$y = \pm \sqrt{4^{2}}$

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

Step 3.12.2.2

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

$y = \pm 4$

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

$y = \pm 4$

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

Step 3.12.3

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

Tap for more steps...

Step 3.12.3.1

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

$y = 4$

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

Step 3.12.3.2

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

$y = - 4$

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

Step 3.12.3.3

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

$y = 4, - 4$

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

$y = 4, - 4$

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

$y = 4, - 4$

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

Step 3.13

Solve the second equation for $y$ .

$y^{2} = 12$

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

Step 3.14

Solve the equation for $y$ .

Tap for more steps...

Step 3.14.1

Remove parentheses.

$y^{2} = 12$

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

Step 3.14.2

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

$y = \pm \sqrt{12}$

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

Step 3.14.3

Simplify $\pm \sqrt{12}$ .

Tap for more steps...

Step 3.14.3.1

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 3.14.3.1.1

Factor $4$ out of $12$ .

$y = \pm \sqrt{4 (3)}$

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

Step 3.14.3.1.2

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

$y = \pm \sqrt{2^{2} \cdot 3}$

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

$y = \pm \sqrt{2^{2} \cdot 3}$

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

Step 3.14.3.2

Pull terms out from under the radical.

$y = \pm 2 \sqrt{3}$

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

$y = \pm 2 \sqrt{3}$

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

Step 3.14.4

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

Tap for more steps...

Step 3.14.4.1

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

$y = 2 \sqrt{3}$

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

Step 3.14.4.2

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

$y = - 2 \sqrt{3}$

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

Step 3.14.4.3

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

$y = 2 \sqrt{3}, - 2 \sqrt{3}$

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

$y = 2 \sqrt{3}, - 2 \sqrt{3}$

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

$y = 2 \sqrt{3}, - 2 \sqrt{3}$

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

Step 3.15

The solution to $\frac{y^{4}}{4} - 7 y^{2} + 64 = 16$ is $y = 4, - 4, 2 \sqrt{3}, - 2 \sqrt{3}$ .

$y = 4, - 4, 2 \sqrt{3}, - 2 \sqrt{3}$

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

$y = 4, - 4, 2 \sqrt{3}, - 2 \sqrt{3}$

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

Step 4

Replace all occurrences of $y$ with $4$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $4$ .

$x = - 8 + \frac{{(4)}^{2}}{2}$

$y = 4$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $- 8 + \frac{{(4)}^{2}}{2}$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Raise $4$ to the power of $2$ .

$x = - 8 + \frac{16}{2}$

$y = 4$

Step 4.2.1.1.2

Divide $16$ by $2$ .

$x = - 8 + 8$

$y = 4$

$x = - 8 + 8$

$y = 4$

Step 4.2.1.2

Add $- 8$ and $8$ .

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

Step 5

Replace all occurrences of $y$ with $- 4$ in each equation.

Tap for more steps...

Step 5.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $- 4$ .

$x = - 8 + \frac{{(- 4)}^{2}}{2}$

$y = - 4$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $- 8 + \frac{{(- 4)}^{2}}{2}$ .

Tap for more steps...

Step 5.2.1.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1.1

Raise $- 4$ to the power of $2$ .

$x = - 8 + \frac{16}{2}$

$y = - 4$

Step 5.2.1.1.2

Divide $16$ by $2$ .

$x = - 8 + 8$

$y = - 4$

$x = - 8 + 8$

$y = - 4$

Step 5.2.1.2

Add $- 8$ and $8$ .

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

Step 6

Replace all occurrences of $y$ with $4$ in each equation.

Tap for more steps...

Step 6.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $4$ .

$x = - 8 + \frac{{(4)}^{2}}{2}$

$y = 4$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Simplify $- 8 + \frac{{(4)}^{2}}{2}$ .

Tap for more steps...

Step 6.2.1.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1.1

Raise $4$ to the power of $2$ .

$x = - 8 + \frac{16}{2}$

$y = 4$

Step 6.2.1.1.2

Divide $16$ by $2$ .

$x = - 8 + 8$

$y = 4$

$x = - 8 + 8$

$y = 4$

Step 6.2.1.2

Add $- 8$ and $8$ .

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

Step 7

Replace all occurrences of $y$ with $- 4$ in each equation.

Tap for more steps...

Step 7.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $- 4$ .

$x = - 8 + \frac{{(- 4)}^{2}}{2}$

$y = - 4$

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

Simplify $- 8 + \frac{{(- 4)}^{2}}{2}$ .

Tap for more steps...

Step 7.2.1.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1.1

Raise $- 4$ to the power of $2$ .

$x = - 8 + \frac{16}{2}$

$y = - 4$

Step 7.2.1.1.2

Divide $16$ by $2$ .

$x = - 8 + 8$

$y = - 4$

$x = - 8 + 8$

$y = - 4$

Step 7.2.1.2

Add $- 8$ and $8$ .

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

Step 8

Replace all occurrences of $y$ with $2 \sqrt{3}$ in each equation.

Tap for more steps...

Step 8.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $2 \sqrt{3}$ .

$x = - 8 + \frac{{(2 \sqrt{3})}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 8.2

Simplify the right side.

Tap for more steps...

Step 8.2.1

Simplify $- 8 + \frac{{(2 \sqrt{3})}^{2}}{2}$ .

Tap for more steps...

Step 8.2.1.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1.1

Simplify the numerator.

Tap for more steps...

Step 8.2.1.1.1.1

Apply the product rule to $2 \sqrt{3}$ .

$x = - 8 + \frac{2^{2} {\sqrt{3}}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.2

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

$x = - 8 + \frac{4 {\sqrt{3}}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.3

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

Tap for more steps...

Step 8.2.1.1.1.3.1

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

$x = - 8 + \frac{4 {(3^{\frac{1}{2}})}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.3.2

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

$x = - 8 + \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.3.3

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

$x = - 8 + \frac{4 \cdot 3^{\frac{2}{2}}}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.1.1.3.4.1

Cancel the common factor.

$x = - 8 + \frac{4 \cdot 3^{\frac{2}{2}}}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.3.4.2

Rewrite the expression.

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.1.3.5

Evaluate the exponent.

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.2

Multiply $4$ by $3$ .

$x = - 8 + \frac{12}{2}$

$y = 2 \sqrt{3}$

Step 8.2.1.1.3

Divide $12$ by $2$ .

$x = - 8 + 6$

$y = 2 \sqrt{3}$

$x = - 8 + 6$

$y = 2 \sqrt{3}$

Step 8.2.1.2

Add $- 8$ and $6$ .

$x = - 2$

$y = 2 \sqrt{3}$

$x = - 2$

$y = 2 \sqrt{3}$

$x = - 2$

$y = 2 \sqrt{3}$

$x = - 2$

$y = 2 \sqrt{3}$

Step 9

Replace all occurrences of $y$ with $4$ in each equation.

Tap for more steps...

Step 9.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $4$ .

$x = - 8 + \frac{{(4)}^{2}}{2}$

$y = 4$

Step 9.2

Simplify the right side.

Tap for more steps...

Step 9.2.1

Simplify $- 8 + \frac{{(4)}^{2}}{2}$ .

Tap for more steps...

Step 9.2.1.1

Simplify each term.

Tap for more steps...

Step 9.2.1.1.1

Raise $4$ to the power of $2$ .

$x = - 8 + \frac{16}{2}$

$y = 4$

Step 9.2.1.1.2

Divide $16$ by $2$ .

$x = - 8 + 8$

$y = 4$

$x = - 8 + 8$

$y = 4$

Step 9.2.1.2

Add $- 8$ and $8$ .

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

Step 10

Replace all occurrences of $y$ with $- 4$ in each equation.

Tap for more steps...

Step 10.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $- 4$ .

$x = - 8 + \frac{{(- 4)}^{2}}{2}$

$y = - 4$

Step 10.2

Simplify the right side.

Tap for more steps...

Step 10.2.1

Simplify $- 8 + \frac{{(- 4)}^{2}}{2}$ .

Tap for more steps...

Step 10.2.1.1

Simplify each term.

Tap for more steps...

Step 10.2.1.1.1

Raise $- 4$ to the power of $2$ .

$x = - 8 + \frac{16}{2}$

$y = - 4$

Step 10.2.1.1.2

Divide $16$ by $2$ .

$x = - 8 + 8$

$y = - 4$

$x = - 8 + 8$

$y = - 4$

Step 10.2.1.2

Add $- 8$ and $8$ .

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

Step 11

Replace all occurrences of $y$ with $2 \sqrt{3}$ in each equation.

Tap for more steps...

Step 11.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $2 \sqrt{3}$ .

$x = - 8 + \frac{{(2 \sqrt{3})}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 11.2

Simplify the right side.

Tap for more steps...

Step 11.2.1

Simplify $- 8 + \frac{{(2 \sqrt{3})}^{2}}{2}$ .

Tap for more steps...

Step 11.2.1.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1.1

Simplify the numerator.

Tap for more steps...

Step 11.2.1.1.1.1

Apply the product rule to $2 \sqrt{3}$ .

$x = - 8 + \frac{2^{2} {\sqrt{3}}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.2

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

$x = - 8 + \frac{4 {\sqrt{3}}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.3

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

Tap for more steps...

Step 11.2.1.1.1.3.1

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

$x = - 8 + \frac{4 {(3^{\frac{1}{2}})}^{2}}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.3.2

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

$x = - 8 + \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.3.3

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

$x = - 8 + \frac{4 \cdot 3^{\frac{2}{2}}}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1.1.1.3.4.1

Cancel the common factor.

$x = - 8 + \frac{4 \cdot 3^{\frac{2}{2}}}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.3.4.2

Rewrite the expression.

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.1.3.5

Evaluate the exponent.

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.2

Multiply $4$ by $3$ .

$x = - 8 + \frac{12}{2}$

$y = 2 \sqrt{3}$

Step 11.2.1.1.3

Divide $12$ by $2$ .

$x = - 8 + 6$

$y = 2 \sqrt{3}$

$x = - 8 + 6$

$y = 2 \sqrt{3}$

Step 11.2.1.2

Add $- 8$ and $6$ .

$x = - 2$

$y = 2 \sqrt{3}$

$x = - 2$

$y = 2 \sqrt{3}$

$x = - 2$

$y = 2 \sqrt{3}$

$x = - 2$

$y = 2 \sqrt{3}$

Step 12

Replace all occurrences of $y$ with $- 2 \sqrt{3}$ in each equation.

Tap for more steps...

Step 12.1

Replace all occurrences of $y$ in $x = - 8 + \frac{y^{2}}{2}$ with $- 2 \sqrt{3}$ .

$x = - 8 + \frac{{(- 2 \sqrt{3})}^{2}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2

Simplify the right side.

Tap for more steps...

Step 12.2.1

Simplify $- 8 + \frac{{(- 2 \sqrt{3})}^{2}}{2}$ .

Tap for more steps...

Step 12.2.1.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1.1

Simplify the numerator.

Tap for more steps...

Step 12.2.1.1.1.1

Apply the product rule to $- 2 \sqrt{3}$ .

$x = - 8 + \frac{{(- 2)}^{2} {\sqrt{3}}^{2}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.2

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

$x = - 8 + \frac{4 {\sqrt{3}}^{2}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.3

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

Tap for more steps...

Step 12.2.1.1.1.3.1

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

$x = - 8 + \frac{4 {(3^{\frac{1}{2}})}^{2}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.3.2

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

$x = - 8 + \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.3.3

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

$x = - 8 + \frac{4 \cdot 3^{\frac{2}{2}}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.2.1.1.1.3.4.1

Cancel the common factor.

$x = - 8 + \frac{4 \cdot 3^{\frac{2}{2}}}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.3.4.2

Rewrite the expression.

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = - 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.1.3.5

Evaluate the exponent.

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = - 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = - 2 \sqrt{3}$

$x = - 8 + \frac{4 \cdot 3}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.2

Multiply $4$ by $3$ .

$x = - 8 + \frac{12}{2}$

$y = - 2 \sqrt{3}$

Step 12.2.1.1.3

Divide $12$ by $2$ .

$x = - 8 + 6$

$y = - 2 \sqrt{3}$

$x = - 8 + 6$

$y = - 2 \sqrt{3}$

Step 12.2.1.2

Add $- 8$ and $6$ .

$x = - 2$

$y = - 2 \sqrt{3}$

$x = - 2$

$y = - 2 \sqrt{3}$

$x = - 2$

$y = - 2 \sqrt{3}$

$x = - 2$

$y = - 2 \sqrt{3}$

Step 13

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

$(0, 4)$

$(0, - 4)$

$(- 2, 2 \sqrt{3})$

$(- 2, - 2 \sqrt{3})$

Step 14

The result can be shown in multiple forms.

Point Form:

$(0, 4), (0, - 4), (- 2, 2 \sqrt{3}), (- 2, - 2 \sqrt{3})$

Equation Form:

$x = 0, y = 4$

$x = 0, y = - 4$

$x = - 2, y = 2 \sqrt{3}$

$x = - 2, y = - 2 \sqrt{3}$

Step 15