Precalculus Examples

$256 x^{2} + 16 y^{2} = 4096$ , $y = x^{2} - 16$

Step 1

Replace all occurrences of $y$ with $x^{2} - 16$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $256 x^{2} + 16 y^{2} = 4096$ with $x^{2} - 16$ .

$256 x^{2} + 16 {(x^{2} - 16)}^{2} = 4096$

$y = x^{2} - 16$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $256 x^{2} + 16 {(x^{2} - 16)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x^{2} - 16)}^{2}$ as $(x^{2} - 16) (x^{2} - 16)$ .

$256 x^{2} + 16 ((x^{2} - 16) (x^{2} - 16)) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.2

Expand $(x^{2} - 16) (x^{2} - 16)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$256 x^{2} + 16 (x^{2} (x^{2} - 16) - 16 (x^{2} - 16)) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.2.2

Apply the distributive property.

$256 x^{2} + 16 (x^{2} x^{2} + x^{2} \cdot - 16 - 16 (x^{2} - 16)) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.2.3

Apply the distributive property.

$256 x^{2} + 16 (x^{2} x^{2} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16) = 4096$

$y = x^{2} - 16$

$256 x^{2} + 16 (x^{2} x^{2} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

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

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Step 1.2.1.1.3.1.1.1

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

$256 x^{2} + 16 (x^{2 + 2} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.3.1.1.2

Add $2$ and $2$ .

$256 x^{2} + 16 (x^{4} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16) = 4096$

$y = x^{2} - 16$

$256 x^{2} + 16 (x^{4} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.3.1.2

Move $- 16$ to the left of $x^{2}$ .

$256 x^{2} + 16 (x^{4} - 16 \cdot x^{2} - 16 x^{2} - 16 \cdot - 16) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.3.1.3

Multiply $- 16$ by $- 16$ .

$256 x^{2} + 16 (x^{4} - 16 x^{2} - 16 x^{2} + 256) = 4096$

$y = x^{2} - 16$

$256 x^{2} + 16 (x^{4} - 16 x^{2} - 16 x^{2} + 256) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.3.2

Subtract $16 x^{2}$ from $- 16 x^{2}$ .

$256 x^{2} + 16 (x^{4} - 32 x^{2} + 256) = 4096$

$y = x^{2} - 16$

$256 x^{2} + 16 (x^{4} - 32 x^{2} + 256) = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.4

Apply the distributive property.

$256 x^{2} + 16 x^{4} + 16 (- 32 x^{2}) + 16 \cdot 256 = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $- 32$ by $16$ .

$256 x^{2} + 16 x^{4} - 512 x^{2} + 16 \cdot 256 = 4096$

$y = x^{2} - 16$

Step 1.2.1.1.5.2

Multiply $16$ by $256$ .

$256 x^{2} + 16 x^{4} - 512 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

$256 x^{2} + 16 x^{4} - 512 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

$256 x^{2} + 16 x^{4} - 512 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

Step 1.2.1.2

Subtract $512 x^{2}$ from $256 x^{2}$ .

$16 x^{4} - 256 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

$16 x^{4} - 256 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

$16 x^{4} - 256 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

$16 x^{4} - 256 x^{2} + 4096 = 4096$

$y = x^{2} - 16$

Step 2

Solve for $x$ in $16 x^{4} - 256 x^{2} + 4096 = 4096$ .

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Step 2.1

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

$16 u^{2} - 256 u + 4096 = 4096$

$u = x^{2}$

$y = x^{2} - 16$

Step 2.2

Subtract $4096$ from both sides of the equation.

$16 u^{2} - 256 u + 4096 - 4096 = 0$

$y = x^{2} - 16$

Step 2.3

Combine the opposite terms in $16 u^{2} - 256 u + 4096 - 4096$ .

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Step 2.3.1

Subtract $4096$ from $4096$ .

$16 u^{2} - 256 u + 0 = 0$

$y = x^{2} - 16$

Step 2.3.2

Add $16 u^{2} - 256 u$ and $0$ .

$16 u^{2} - 256 u = 0$

$y = x^{2} - 16$

$16 u^{2} - 256 u = 0$

$y = x^{2} - 16$

Step 2.4

Factor $16 u$ out of $16 u^{2} - 256 u$ .

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Step 2.4.1

Factor $16 u$ out of $16 u^{2}$ .

$16 u (u) - 256 u = 0$

$y = x^{2} - 16$

Step 2.4.2

Factor $16 u$ out of $- 256 u$ .

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

$y = x^{2} - 16$

Step 2.4.3

Factor $16 u$ out of $16 u (u) + 16 u (- 16)$ .

$16 u (u - 16) = 0$

$y = x^{2} - 16$

$16 u (u - 16) = 0$

$y = x^{2} - 16$

Step 2.5

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

$u = 0$

$u - 16 = 0$

$y = x^{2} - 16$

Step 2.6

Set $u$ equal to $0$ .

$u = 0$

$y = x^{2} - 16$

Step 2.7

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

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Step 2.7.1

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

$u - 16 = 0$

$y = x^{2} - 16$

Step 2.7.2

Add $16$ to both sides of the equation.

$u = 16$

$y = x^{2} - 16$

$u = 16$

$y = x^{2} - 16$

Step 2.8

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

$u = 0, 16$

$y = x^{2} - 16$

Step 2.9

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

$x^{2} = 0$

$x^{2} = 16$

$y = x^{2} - 16$

Step 2.10

Solve the first equation for $x$ .

$x^{2} = 0$

$y = x^{2} - 16$

Step 2.11

Solve the equation for $x$ .

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Step 2.11.1

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

$x = \pm \sqrt{0}$

$y = x^{2} - 16$

Step 2.11.2

Simplify $\pm \sqrt{0}$ .

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Step 2.11.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

$y = x^{2} - 16$

Step 2.11.2.2

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

$x = \pm 0$

$y = x^{2} - 16$

Step 2.11.2.3

Plus or minus $0$ is $0$ .

$x = 0$

$y = x^{2} - 16$

$x = 0$

$y = x^{2} - 16$

$x = 0$

$y = x^{2} - 16$

Step 2.12

Solve the second equation for $x$ .

$x^{2} = 16$

$y = x^{2} - 16$

Step 2.13

Solve the equation for $x$ .

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Step 2.13.1

Remove parentheses.

$x^{2} = 16$

$y = x^{2} - 16$

Step 2.13.2

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

$x = \pm \sqrt{16}$

$y = x^{2} - 16$

Step 2.13.3

Simplify $\pm \sqrt{16}$ .

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Step 2.13.3.1

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

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

$y = x^{2} - 16$

Step 2.13.3.2

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

$x = \pm 4$

$y = x^{2} - 16$

$x = \pm 4$

$y = x^{2} - 16$

Step 2.13.4

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

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Step 2.13.4.1

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

$x = 4$

$y = x^{2} - 16$

Step 2.13.4.2

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

$x = - 4$

$y = x^{2} - 16$

Step 2.13.4.3

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

$x = 4, - 4$

$y = x^{2} - 16$

$x = 4, - 4$

$y = x^{2} - 16$

$x = 4, - 4$

$y = x^{2} - 16$

Step 2.14

The solution to $16 x^{4} - 256 x^{2} + 4096 = 4096$ is $x = 0, 4, - 4$ .

$x = 0, 4, - 4$

$y = x^{2} - 16$

$x = 0, 4, - 4$

$y = x^{2} - 16$

Step 3

Replace all occurrences of $x$ with $0$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} - 16$ with $0$ .

$y = {(0)}^{2} - 16$

$x = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(0)}^{2} - 16$ .

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Step 3.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 16$

$x = 0$

Step 3.2.1.2

Subtract $16$ from $0$ .

$y = - 16$

$x = 0$

$y = - 16$

$x = 0$

$y = - 16$

$x = 0$

$y = - 16$

$x = 0$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} - 16$ with $4$ .

$y = {(4)}^{2} - 16$

$x = 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(4)}^{2} - 16$ .

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Step 4.2.1.1

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

$y = 16 - 16$

$x = 4$

Step 4.2.1.2

Subtract $16$ from $16$ .

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

Step 5

Replace all occurrences of $x$ with $0$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = x^{2} - 16$ with $0$ .

$y = {(0)}^{2} - 16$

$x = 0$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify ${(0)}^{2} - 16$ .

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Step 5.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 16$

$x = 0$

Step 5.2.1.2

Subtract $16$ from $0$ .

$y = - 16$

$x = 0$

$y = - 16$

$x = 0$

$y = - 16$

$x = 0$

$y = - 16$

$x = 0$

Step 6

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

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Step 6.1

Replace all occurrences of $x$ in $y = x^{2} - 16$ with $4$ .

$y = {(4)}^{2} - 16$

$x = 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${(4)}^{2} - 16$ .

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Step 6.2.1.1

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

$y = 16 - 16$

$x = 4$

Step 6.2.1.2

Subtract $16$ from $16$ .

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

Step 7

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

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Step 7.1

Replace all occurrences of $x$ in $y = x^{2} - 16$ with $- 4$ .

$y = {(- 4)}^{2} - 16$

$x = - 4$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify ${(- 4)}^{2} - 16$ .

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Step 7.2.1.1

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

$y = 16 - 16$

$x = - 4$

Step 7.2.1.2

Subtract $16$ from $16$ .

$y = 0$

$x = - 4$

$y = 0$

$x = - 4$

$y = 0$

$x = - 4$

$y = 0$

$x = - 4$

Step 8

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

$(0, - 16)$

$(4, 0)$

$(- 4, 0)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(0, - 16), (4, 0), (- 4, 0)$

Equation Form:

$x = 0, y = - 16$

$x = 4, y = 0$

$x = - 4, y = 0$

Step 10