Precalculus Examples

$x^{2} + y^{2} - 16 y + 39 = 0$ , $y^{2} - x^{2} - 9 = 0$

Step 1

Solve for $y$ in $y^{2} - x^{2} - 9 = 0$ .

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Step 1.1

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

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Step 1.1.1

Add $x^{2}$ to both sides of the equation.

$y^{2} - 9 = x^{2}$

$x^{2} + y^{2} - 16 y + 39 = 0$

Step 1.1.2

Add $9$ to both sides of the equation.

$y^{2} = x^{2} + 9$

$x^{2} + y^{2} - 16 y + 39 = 0$

$y^{2} = x^{2} + 9$

$x^{2} + y^{2} - 16 y + 39 = 0$

Step 1.2

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

$y = \pm \sqrt{x^{2} + 9}$

$x^{2} + y^{2} - 16 y + 39 = 0$

Step 1.3

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

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Step 1.3.1

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

$y = \sqrt{x^{2} + 9}$

$x^{2} + y^{2} - 16 y + 39 = 0$

Step 1.3.2

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

$y = - \sqrt{x^{2} + 9}$

$x^{2} + y^{2} - 16 y + 39 = 0$

Step 1.3.3

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

$y = \sqrt{x^{2} + 9}$

$y = - \sqrt{x^{2} + 9}$

$x^{2} + y^{2} - 16 y + 39 = 0$

$y = \sqrt{x^{2} + 9}$

$y = - \sqrt{x^{2} + 9}$

$x^{2} + y^{2} - 16 y + 39 = 0$

$y = \sqrt{x^{2} + 9}$

$y = - \sqrt{x^{2} + 9}$

$x^{2} + y^{2} - 16 y + 39 = 0$

Step 2

Solve the system $y = \sqrt{x^{2} + 9}, x^{2} + y^{2} - 16 y + 39 = 0$ .

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Step 2.1

Replace all occurrences of $y$ with $\sqrt{x^{2} + 9}$ in each equation.

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Step 2.1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} - 16 y + 39 = 0$ with $\sqrt{x^{2} + 9}$ .

$x^{2} + {(\sqrt{x^{2} + 9})}^{2} - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $x^{2} + {(\sqrt{x^{2} + 9})}^{2} - 16 \sqrt{x^{2} + 9} + 39$ .

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Step 2.1.2.1.1

Rewrite ${\sqrt{x^{2} + 9}}^{2}$ as $x^{2} + 9$ .

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Step 2.1.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 9}$ as ${(x^{2} + 9)}^{\frac{1}{2}}$ .

$x^{2} + {({(x^{2} + 9)}^{\frac{1}{2}})}^{2} - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.1.2

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

$x^{2} + {(x^{2} + 9)}^{\frac{1}{2} \cdot 2} - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.1.3

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

$x^{2} + {(x^{2} + 9)}^{\frac{2}{2}} - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.4.1

Cancel the common factor.

$x^{2} + {(x^{2} + 9)}^{\frac{2}{2}} - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.1.4.2

Rewrite the expression.

$x^{2} + x^{2} + 9 - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

$x^{2} + x^{2} + 9 - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.1.5

Simplify.

$x^{2} + x^{2} + 9 - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

$x^{2} + x^{2} + 9 - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.2

Simplify by adding terms.

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Step 2.1.2.1.2.1

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 9 - 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.1.2.1.2.2

Add $9$ and $39$ .

$2 x^{2} + 48 - 16 \sqrt{x^{2} + 9} = 0$

$y = \sqrt{x^{2} + 9}$

$2 x^{2} + 48 - 16 \sqrt{x^{2} + 9} = 0$

$y = \sqrt{x^{2} + 9}$

$2 x^{2} + 48 - 16 \sqrt{x^{2} + 9} = 0$

$y = \sqrt{x^{2} + 9}$

$2 x^{2} + 48 - 16 \sqrt{x^{2} + 9} = 0$

$y = \sqrt{x^{2} + 9}$

$2 x^{2} + 48 - 16 \sqrt{x^{2} + 9} = 0$

$y = \sqrt{x^{2} + 9}$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 4, 0, 4$

$y = \sqrt{x^{2} + 9}$

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 9}$ with $- 4$ .

$y = \sqrt{{(- 4)}^{2} + 9}$

$x = - 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{{(- 4)}^{2} + 9}$ .

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Step 2.3.2.1.1

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

$y = \sqrt{16 + 9}$

$x = - 4$

Step 2.3.2.1.2

Add $16$ and $9$ .

$y = \sqrt{25}$

$x = - 4$

Step 2.3.2.1.3

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

$y = \sqrt{5^{2}}$

$x = - 4$

Step 2.3.2.1.4

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

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

Step 2.4

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

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Step 2.4.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 9}$ with $0$ .

$y = \sqrt{{(0)}^{2} + 9}$

$x = 0$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{{(0)}^{2} + 9}$ .

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Step 2.4.2.1.1

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

$y = \sqrt{0 + 9}$

$x = 0$

Step 2.4.2.1.2

Add $0$ and $9$ .

$y = \sqrt{9}$

$x = 0$

Step 2.4.2.1.3

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

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

$x = 0$

Step 2.4.2.1.4

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

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

Step 2.5

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

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Step 2.5.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 9}$ with $- 4$ .

$y = \sqrt{{(- 4)}^{2} + 9}$

$x = - 4$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $\sqrt{{(- 4)}^{2} + 9}$ .

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Step 2.5.2.1.1

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

$y = \sqrt{16 + 9}$

$x = - 4$

Step 2.5.2.1.2

Add $16$ and $9$ .

$y = \sqrt{25}$

$x = - 4$

Step 2.5.2.1.3

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

$y = \sqrt{5^{2}}$

$x = - 4$

Step 2.5.2.1.4

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

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

$y = 5$

$x = - 4$

Step 2.6

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

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Step 2.6.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 9}$ with $0$ .

$y = \sqrt{{(0)}^{2} + 9}$

$x = 0$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $\sqrt{{(0)}^{2} + 9}$ .

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Step 2.6.2.1.1

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

$y = \sqrt{0 + 9}$

$x = 0$

Step 2.6.2.1.2

Add $0$ and $9$ .

$y = \sqrt{9}$

$x = 0$

Step 2.6.2.1.3

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

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

$x = 0$

Step 2.6.2.1.4

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

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

Step 2.7

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

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Step 2.7.1

Replace all occurrences of $x$ in $y = \sqrt{x^{2} + 9}$ with $4$ .

$y = \sqrt{{(4)}^{2} + 9}$

$x = 4$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Simplify $\sqrt{{(4)}^{2} + 9}$ .

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Step 2.7.2.1.1

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

$y = \sqrt{16 + 9}$

$x = 4$

Step 2.7.2.1.2

Add $16$ and $9$ .

$y = \sqrt{25}$

$x = 4$

Step 2.7.2.1.3

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

$y = \sqrt{5^{2}}$

$x = 4$

Step 2.7.2.1.4

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

$y = 5$

$x = 4$

$y = 5$

$x = 4$

$y = 5$

$x = 4$

$y = 5$

$x = 4$

$y = 5$

$x = 4$

Step 3

Solve the system $y = - \sqrt{x^{2} + 9}, x^{2} + y^{2} - 16 y + 39 = 0$ .

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Step 3.1

Replace all occurrences of $y$ with $- \sqrt{x^{2} + 9}$ in each equation.

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Step 3.1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} - 16 y + 39 = 0$ with $- \sqrt{x^{2} + 9}$ .

$x^{2} + {(- \sqrt{x^{2} + 9})}^{2} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $x^{2} + {(- \sqrt{x^{2} + 9})}^{2} - 16 (- \sqrt{x^{2} + 9}) + 39$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{x^{2} + 9}$ .

$x^{2} + {(- 1)}^{2} {\sqrt{x^{2} + 9}}^{2} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} + 1 {\sqrt{x^{2} + 9}}^{2} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{x^{2} + 9}}^{2}$ by $1$ .

$x^{2} + {\sqrt{x^{2} + 9}}^{2} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{x^{2} + 9}}^{2}$ as $x^{2} + 9$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 9}$ as ${(x^{2} + 9)}^{\frac{1}{2}}$ .

$x^{2} + {({(x^{2} + 9)}^{\frac{1}{2}})}^{2} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.4.2

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

$x^{2} + {(x^{2} + 9)}^{\frac{1}{2} \cdot 2} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.4.3

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

$x^{2} + {(x^{2} + 9)}^{\frac{2}{2}} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

$x^{2} + {(x^{2} + 9)}^{\frac{2}{2}} - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$x^{2} + x^{2} + 9 - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

$x^{2} + x^{2} + 9 - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.4.5

Simplify.

$x^{2} + x^{2} + 9 - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

$x^{2} + x^{2} + 9 - 16 (- \sqrt{x^{2} + 9}) + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.1.5

Multiply $- 1$ by $- 16$ .

$x^{2} + x^{2} + 9 + 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

$x^{2} + x^{2} + 9 + 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.2

Simplify by adding terms.

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Step 3.1.2.1.2.1

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 9 + 16 \sqrt{x^{2} + 9} + 39 = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.1.2.1.2.2

Add $9$ and $39$ .

$2 x^{2} + 48 + 16 \sqrt{x^{2} + 9} = 0$

$y = - \sqrt{x^{2} + 9}$

$2 x^{2} + 48 + 16 \sqrt{x^{2} + 9} = 0$

$y = - \sqrt{x^{2} + 9}$

$2 x^{2} + 48 + 16 \sqrt{x^{2} + 9} = 0$

$y = - \sqrt{x^{2} + 9}$

$2 x^{2} + 48 + 16 \sqrt{x^{2} + 9} = 0$

$y = - \sqrt{x^{2} + 9}$

$2 x^{2} + 48 + 16 \sqrt{x^{2} + 9} = 0$

$y = - \sqrt{x^{2} + 9}$

Step 3.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

$y = - \sqrt{x^{2} + 9}$

No solution

Step 4