Precalculus Examples

$y = x^{2} - 13$ , $x^{2} + y^{2} = 25$

Step 1

Replace all occurrences of $y$ with $x^{2} - 13$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 25$ with $x^{2} - 13$ .

$x^{2} + {(x^{2} - 13)}^{2} = 25$

$y = x^{2} - 13$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $x^{2} + {(x^{2} - 13)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x^{2} - 13)}^{2}$ as $(x^{2} - 13) (x^{2} - 13)$ .

$x^{2} + (x^{2} - 13) (x^{2} - 13) = 25$

$y = x^{2} - 13$

Step 1.2.1.1.2

Expand $(x^{2} - 13) (x^{2} - 13)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$x^{2} + x^{2} (x^{2} - 13) - 13 (x^{2} - 13) = 25$

$y = x^{2} - 13$

Step 1.2.1.1.2.2

Apply the distributive property.

$x^{2} + x^{2} x^{2} + x^{2} \cdot - 13 - 13 (x^{2} - 13) = 25$

$y = x^{2} - 13$

Step 1.2.1.1.2.3

Apply the distributive property.

$x^{2} + x^{2} x^{2} + x^{2} \cdot - 13 - 13 x^{2} - 13 \cdot - 13 = 25$

$y = x^{2} - 13$

$x^{2} + x^{2} x^{2} + x^{2} \cdot - 13 - 13 x^{2} - 13 \cdot - 13 = 25$

$y = x^{2} - 13$

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.

$x^{2} + x^{2 + 2} + x^{2} \cdot - 13 - 13 x^{2} - 13 \cdot - 13 = 25$

$y = x^{2} - 13$

Step 1.2.1.1.3.1.1.2

Add $2$ and $2$ .

$x^{2} + x^{4} + x^{2} \cdot - 13 - 13 x^{2} - 13 \cdot - 13 = 25$

$y = x^{2} - 13$

$x^{2} + x^{4} + x^{2} \cdot - 13 - 13 x^{2} - 13 \cdot - 13 = 25$

$y = x^{2} - 13$

Step 1.2.1.1.3.1.2

Move $- 13$ to the left of $x^{2}$ .

$x^{2} + x^{4} - 13 \cdot x^{2} - 13 x^{2} - 13 \cdot - 13 = 25$

$y = x^{2} - 13$

Step 1.2.1.1.3.1.3

Multiply $- 13$ by $- 13$ .

$x^{2} + x^{4} - 13 x^{2} - 13 x^{2} + 169 = 25$

$y = x^{2} - 13$

$x^{2} + x^{4} - 13 x^{2} - 13 x^{2} + 169 = 25$

$y = x^{2} - 13$

Step 1.2.1.1.3.2

Subtract $13 x^{2}$ from $- 13 x^{2}$ .

$x^{2} + x^{4} - 26 x^{2} + 169 = 25$

$y = x^{2} - 13$

$x^{2} + x^{4} - 26 x^{2} + 169 = 25$

$y = x^{2} - 13$

$x^{2} + x^{4} - 26 x^{2} + 169 = 25$

$y = x^{2} - 13$

Step 1.2.1.2

Subtract $26 x^{2}$ from $x^{2}$ .

$x^{4} - 25 x^{2} + 169 = 25$

$y = x^{2} - 13$

$x^{4} - 25 x^{2} + 169 = 25$

$y = x^{2} - 13$

$x^{4} - 25 x^{2} + 169 = 25$

$y = x^{2} - 13$

$x^{4} - 25 x^{2} + 169 = 25$

$y = x^{2} - 13$

Step 2

Solve for $x$ in $x^{4} - 25 x^{2} + 169 = 25$ .

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Step 2.1

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

$u^{2} - 25 u + 169 = 25$

$u = x^{2}$

$y = x^{2} - 13$

Step 2.2

Subtract $25$ from both sides of the equation.

$u^{2} - 25 u + 169 - 25 = 0$

$y = x^{2} - 13$

Step 2.3

Subtract $25$ from $169$ .

$u^{2} - 25 u + 144 = 0$

$y = x^{2} - 13$

Step 2.4

Factor $u^{2} - 25 u + 144$ using the AC method.

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Step 2.4.1

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 $144$ and whose sum is $- 25$ .

$- 16, - 9$

$y = x^{2} - 13$

Step 2.4.2

Write the factored form using these integers.

$(u - 16) (u - 9) = 0$

$y = x^{2} - 13$

$(u - 16) (u - 9) = 0$

$y = x^{2} - 13$

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 $0$ .

$u - 16 = 0$

$u - 9 = 0$

$y = x^{2} - 13$

Step 2.6

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

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Step 2.6.1

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

$u - 16 = 0$

$y = x^{2} - 13$

Step 2.6.2

Add $16$ to both sides of the equation.

$u = 16$

$y = x^{2} - 13$

$u = 16$

$y = x^{2} - 13$

Step 2.7

Set $u - 9$ equal to $0$ and solve for $u$ .

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Step 2.7.1

Set $u - 9$ equal to $0$ .

$u - 9 = 0$

$y = x^{2} - 13$

Step 2.7.2

Add $9$ to both sides of the equation.

$u = 9$

$y = x^{2} - 13$

$u = 9$

$y = x^{2} - 13$

Step 2.8

The final solution is all the values that make $(u - 16) (u - 9) = 0$ true.

$u = 16, 9$

$y = x^{2} - 13$

Step 2.9

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

$x^{2} = 16$

$x^{2} = 9$

$y = x^{2} - 13$

Step 2.10

Solve the first equation for $x$ .

$x^{2} = 16$

$y = x^{2} - 13$

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.

$x = \pm \sqrt{16}$

$y = x^{2} - 13$

Step 2.11.2

Simplify $\pm \sqrt{16}$ .

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Step 2.11.2.1

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

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

$y = x^{2} - 13$

Step 2.11.2.2

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

$x = \pm 4$

$y = x^{2} - 13$

$x = \pm 4$

$y = x^{2} - 13$

Step 2.11.3

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

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Step 2.11.3.1

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

$x = 4$

$y = x^{2} - 13$

Step 2.11.3.2

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

$x = - 4$

$y = x^{2} - 13$

Step 2.11.3.3

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

$x = 4, - 4$

$y = x^{2} - 13$

$x = 4, - 4$

$y = x^{2} - 13$

$x = 4, - 4$

$y = x^{2} - 13$

Step 2.12

Solve the second equation for $x$ .

$x^{2} = 9$

$y = x^{2} - 13$

Step 2.13

Solve the equation for $x$ .

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Step 2.13.1

Remove parentheses.

$x^{2} = 9$

$y = x^{2} - 13$

Step 2.13.2

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

$x = \pm \sqrt{9}$

$y = x^{2} - 13$

Step 2.13.3

Simplify $\pm \sqrt{9}$ .

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Step 2.13.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

$y = x^{2} - 13$

Step 2.13.3.2

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

$x = \pm 3$

$y = x^{2} - 13$

$x = \pm 3$

$y = x^{2} - 13$

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.

$x = 3$

$y = x^{2} - 13$

Step 2.13.4.2

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

$x = - 3$

$y = x^{2} - 13$

Step 2.13.4.3

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

$x = 3, - 3$

$y = x^{2} - 13$

$x = 3, - 3$

$y = x^{2} - 13$

$x = 3, - 3$

$y = x^{2} - 13$

Step 2.14

The solution to $x^{4} - 25 x^{2} + 169 = 25$ is $x = 4, - 4, 3, - 3$ .

$x = 4, - 4, 3, - 3$

$y = x^{2} - 13$

$x = 4, - 4, 3, - 3$

$y = x^{2} - 13$

Step 3

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

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $4$ .

$y = {(4)}^{2} - 13$

$x = 4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(4)}^{2} - 13$ .

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Step 3.2.1.1

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

$y = 16 - 13$

$x = 4$

Step 3.2.1.2

Subtract $13$ from $16$ .

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

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 $y = x^{2} - 13$ with $- 4$ .

$y = {(- 4)}^{2} - 13$

$x = - 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(- 4)}^{2} - 13$ .

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Step 4.2.1.1

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

$y = 16 - 13$

$x = - 4$

Step 4.2.1.2

Subtract $13$ from $16$ .

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

Step 5

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

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Step 5.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $4$ .

$y = {(4)}^{2} - 13$

$x = 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify ${(4)}^{2} - 13$ .

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Step 5.2.1.1

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

$y = 16 - 13$

$x = 4$

Step 5.2.1.2

Subtract $13$ from $16$ .

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

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 $y = x^{2} - 13$ with $- 4$ .

$y = {(- 4)}^{2} - 13$

$x = - 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${(- 4)}^{2} - 13$ .

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Step 6.2.1.1

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

$y = 16 - 13$

$x = - 4$

Step 6.2.1.2

Subtract $13$ from $16$ .

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

Step 7

Replace all occurrences of $x$ with $3$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $3$ .

$y = {(3)}^{2} - 13$

$x = 3$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify ${(3)}^{2} - 13$ .

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Step 7.2.1.1

Raise $3$ to the power of $2$ .

$y = 9 - 13$

$x = 3$

Step 7.2.1.2

Subtract $13$ from $9$ .

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

Step 8

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

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Step 8.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $4$ .

$y = {(4)}^{2} - 13$

$x = 4$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify ${(4)}^{2} - 13$ .

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Step 8.2.1.1

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

$y = 16 - 13$

$x = 4$

Step 8.2.1.2

Subtract $13$ from $16$ .

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

Step 9

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

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Step 9.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $- 4$ .

$y = {(- 4)}^{2} - 13$

$x = - 4$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify ${(- 4)}^{2} - 13$ .

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Step 9.2.1.1

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

$y = 16 - 13$

$x = - 4$

Step 9.2.1.2

Subtract $13$ from $16$ .

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

$y = 3$

$x = - 4$

Step 10

Replace all occurrences of $x$ with $3$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $3$ .

$y = {(3)}^{2} - 13$

$x = 3$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify ${(3)}^{2} - 13$ .

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Step 10.2.1.1

Raise $3$ to the power of $2$ .

$y = 9 - 13$

$x = 3$

Step 10.2.1.2

Subtract $13$ from $9$ .

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

Step 11

Replace all occurrences of $x$ with $- 3$ in each equation.

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Step 11.1

Replace all occurrences of $x$ in $y = x^{2} - 13$ with $- 3$ .

$y = {(- 3)}^{2} - 13$

$x = - 3$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify ${(- 3)}^{2} - 13$ .

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Step 11.2.1.1

Raise $- 3$ to the power of $2$ .

$y = 9 - 13$

$x = - 3$

Step 11.2.1.2

Subtract $13$ from $9$ .

$y = - 4$

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

$x = - 3$

Step 12

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

$(4, 3)$

$(- 4, 3)$

$(3, - 4)$

$(- 3, - 4)$

Step 13

The result can be shown in multiple forms.

Point Form:

$(4, 3), (- 4, 3), (3, - 4), (- 3, - 4)$

Equation Form:

$x = 4, y = 3$

$x = - 4, y = 3$

$x = 3, y = - 4$

$x = - 3, y = - 4$

Step 14