Precalculus Examples

$x^{2} - 4 y = 19$ , $x^{2} + y^{2} = 16$

Step 1

Solve for $y$ in $x^{2} - 4 y = 19$ .

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Step 1.1

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

$- 4 y = 19 - x^{2}$

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

Step 1.2

Divide each term in $- 4 y = 19 - x^{2}$ by $- 4$ and simplify.

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Step 1.2.1

Divide each term in $- 4 y = 19 - x^{2}$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{19}{- 4} + \frac{- x^{2}}{- 4}$

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{19}{- 4} + \frac{- x^{2}}{- 4}$

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{19}{- 4} + \frac{- x^{2}}{- 4}$

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

$y = \frac{19}{- 4} + \frac{- x^{2}}{- 4}$

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

$y = \frac{19}{- 4} + \frac{- x^{2}}{- 4}$

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{19}{4} + \frac{- x^{2}}{- 4}$

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

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

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

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

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

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

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

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

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

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

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

Step 2

Replace all occurrences of $y$ with $- \frac{19}{4} + \frac{x^{2}}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 16$ with $- \frac{19}{4} + \frac{x^{2}}{4}$ .

$x^{2} + {(- \frac{19}{4} + \frac{x^{2}}{4})}^{2} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(- \frac{19}{4} + \frac{x^{2}}{4})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- \frac{19}{4} + \frac{x^{2}}{4})}^{2}$ as $(- \frac{19}{4} + \frac{x^{2}}{4}) (- \frac{19}{4} + \frac{x^{2}}{4})$ .

$x^{2} + (- \frac{19}{4} + \frac{x^{2}}{4}) (- \frac{19}{4} + \frac{x^{2}}{4}) = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.2

Expand $(- \frac{19}{4} + \frac{x^{2}}{4}) (- \frac{19}{4} + \frac{x^{2}}{4})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} - \frac{19}{4} \cdot (- \frac{19}{4} + \frac{x^{2}}{4}) + \frac{x^{2}}{4} \cdot (- \frac{19}{4} + \frac{x^{2}}{4}) = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} - \frac{19}{4} \cdot (- \frac{19}{4}) - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4} + \frac{x^{2}}{4}) = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} - \frac{19}{4} \cdot (- \frac{19}{4}) - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} - \frac{19}{4} \cdot (- \frac{19}{4}) - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $- \frac{19}{4} (- \frac{19}{4})$ .

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Step 2.2.1.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$x^{2} + 1 (\frac{19}{4}) \cdot \frac{19}{4} - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.1.2

Multiply $\frac{19}{4}$ by $1$ .

$x^{2} + \frac{19}{4} \cdot \frac{19}{4} - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.1.3

Multiply $\frac{19}{4}$ by $\frac{19}{4}$ .

$x^{2} + \frac{19 \cdot 19}{4 \cdot 4} - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.1.4

Multiply $19$ by $19$ .

$x^{2} + \frac{361}{4 \cdot 4} - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.1.5

Multiply $4$ by $4$ .

$x^{2} + \frac{361}{16} - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{19}{4} \cdot \frac{x^{2}}{4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.2

Multiply $- \frac{19}{4} \cdot \frac{x^{2}}{4}$ .

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Step 2.2.1.1.3.1.2.1

Multiply $\frac{x^{2}}{4}$ by $\frac{19}{4}$ .

$x^{2} + \frac{361}{16} - \frac{x^{2} \cdot 19}{4 \cdot 4} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.2.2

Multiply $4$ by $4$ .

$x^{2} + \frac{361}{16} - \frac{x^{2} \cdot 19}{16} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{x^{2} \cdot 19}{16} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.3

Move $19$ to the left of $x^{2}$ .

$x^{2} + \frac{361}{16} - \frac{19 \cdot x^{2}}{16} + \frac{x^{2}}{4} \cdot (- \frac{19}{4}) + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.4

Multiply $\frac{x^{2}}{4} (- \frac{19}{4})$ .

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Step 2.2.1.1.3.1.4.1

Multiply $\frac{x^{2}}{4}$ by $\frac{19}{4}$ .

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{x^{2} \cdot 19}{4 \cdot 4} + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.4.2

Multiply $4$ by $4$ .

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{x^{2} \cdot 19}{16} + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{x^{2} \cdot 19}{16} + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.5

Move $19$ to the left of $x^{2}$ .

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 \cdot x^{2}}{16} + \frac{x^{2}}{4} \cdot \frac{x^{2}}{4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.6

Combine.

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 x^{2}}{16} + \frac{x^{2} x^{2}}{4 \cdot 4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.7

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

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Step 2.2.1.1.3.1.7.1

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

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 x^{2}}{16} + \frac{x^{2 + 2}}{4 \cdot 4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.7.2

Add $2$ and $2$ .

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 x^{2}}{16} + \frac{x^{4}}{4 \cdot 4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 x^{2}}{16} + \frac{x^{4}}{4 \cdot 4} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.1.8

Multiply $4$ by $4$ .

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 x^{2}}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{16} - \frac{19 x^{2}}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.3.2

Subtract $\frac{19 x^{2}}{16}$ from $- \frac{19 x^{2}}{16}$ .

$x^{2} + \frac{361}{16} - 2 \frac{19 x^{2}}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - 2 \frac{19 x^{2}}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1.1

Factor $2$ out of $- 2$ .

$x^{2} + \frac{361}{16} + 2 (- 1) (\frac{19 x^{2}}{16}) + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.4.1.2

Factor $2$ out of $16$ .

$x^{2} + \frac{361}{16} + 2 \cdot (- 1 \frac{19 x^{2}}{2 \cdot 8}) + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.4.1.3

Cancel the common factor.

$x^{2} + \frac{361}{16} + 2 \cdot (- 1 \frac{19 x^{2}}{2 \cdot 8}) + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.4.1.4

Rewrite the expression.

$x^{2} + \frac{361}{16} - 1 \frac{19 x^{2}}{8} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - 1 \frac{19 x^{2}}{8} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.1.4.2

Rewrite $- 1 \frac{19 x^{2}}{8}$ as $- \frac{19 x^{2}}{8}$ .

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{8} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{8} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x^{2} + \frac{361}{16} - \frac{19 x^{2}}{8} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} \cdot \frac{8}{8} - \frac{19 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.3

Combine $x^{2}$ and $\frac{8}{8}$ .

$\frac{x^{2} \cdot 8}{8} - \frac{19 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 8 - 19 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.5

Subtract $19 x^{2}$ from $x^{2} \cdot 8$ .

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Step 2.2.1.5.1

Reorder $x^{2}$ and $8$ .

$\frac{8 \cdot x^{2} - 19 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.5.2

Subtract $19 x^{2}$ from $8 \cdot x^{2}$ .

$\frac{- 11 \cdot x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$\frac{- 11 \cdot x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 2.2.1.6

Move the negative in front of the fraction.

$- \frac{11 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$- \frac{11 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$- \frac{11 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$- \frac{11 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3

Solve for $x$ in $- \frac{11 x^{2}}{8} + \frac{361}{16} + \frac{x^{4}}{16} = 16$ .

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Step 3.1

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

$\frac{u^{2}}{16} - \frac{11 u}{8} + \frac{361}{16} = 16$

$u = x^{2}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2

Multiply each term in $\frac{u^{2}}{16} - \frac{11 u}{8} + \frac{361}{16} = 16$ by $16$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{u^{2}}{16} - \frac{11 u}{8} + \frac{361}{16} = 16$ by $16$ .

$\frac{u^{2}}{16} \cdot 16 - \frac{11 u}{8} \cdot 16 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $16$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{u^{2}}{16} \cdot 16 - \frac{11 u}{8} \cdot 16 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.1.2

Rewrite the expression.

$u^{2} - \frac{11 u}{8} \cdot 16 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - \frac{11 u}{8} \cdot 16 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.2

Cancel the common factor of $8$ .

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Step 3.2.2.1.2.1

Move the leading negative in $- \frac{11 u}{8}$ into the numerator.

$u^{2} + \frac{- 11 u}{8} \cdot 16 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.2.2

Factor $8$ out of $16$ .

$u^{2} + \frac{- 11 u}{8} \cdot (8 (2)) + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.2.3

Cancel the common factor.

$u^{2} + \frac{- 11 u}{8} \cdot (8 \cdot 2) + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.2.4

Rewrite the expression.

$u^{2} - 11 u \cdot 2 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - 11 u \cdot 2 + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.3

Multiply $2$ by $- 11$ .

$u^{2} - 22 u + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.4

Cancel the common factor of $16$ .

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Step 3.2.2.1.4.1

Cancel the common factor.

$u^{2} - 22 u + \frac{361}{16} \cdot 16 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.2.1.4.2

Rewrite the expression.

$u^{2} - 22 u + 361 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - 22 u + 361 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - 22 u + 361 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - 22 u + 361 = 16 \cdot 16$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply $16$ by $16$ .

$u^{2} - 22 u + 361 = 256$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - 22 u + 361 = 256$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u^{2} - 22 u + 361 = 256$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.3

Subtract $256$ from both sides of the equation.

$u^{2} - 22 u + 361 - 256 = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.4

Subtract $256$ from $361$ .

$u^{2} - 22 u + 105 = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.5

Factor $u^{2} - 22 u + 105$ using the AC method.

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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 $105$ and whose sum is $- 22$ .

$- 15, - 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.5.2

Write the factored form using these integers.

$(u - 15) (u - 7) = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$(u - 15) (u - 7) = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

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 $0$ .

$u - 15 = 0$

$u - 7 = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.7

Set $u - 15$ equal to $0$ and solve for $u$ .

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Step 3.7.1

Set $u - 15$ equal to $0$ .

$u - 15 = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.7.2

Add $15$ to both sides of the equation.

$u = 15$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u = 15$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.8

Set $u - 7$ equal to $0$ and solve for $u$ .

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Step 3.8.1

Set $u - 7$ equal to $0$ .

$u - 7 = 0$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.8.2

Add $7$ to both sides of the equation.

$u = 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$u = 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.9

The final solution is all the values that make $(u - 15) (u - 7) = 0$ true.

$u = 15, 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.10

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

$x^{2} = 15$

$x^{2} = 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.11

Solve the first equation for $x$ .

$x^{2} = 15$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.12

Solve the equation for $x$ .

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Step 3.12.1

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

$x = \pm \sqrt{15}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.12.2

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

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Step 3.12.2.1

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

$x = \sqrt{15}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.12.2.2

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

$x = - \sqrt{15}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.12.2.3

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

$x = \sqrt{15}, - \sqrt{15}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x = \sqrt{15}, - \sqrt{15}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x = \sqrt{15}, - \sqrt{15}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.13

Solve the second equation for $x$ .

$x^{2} = 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.14

Solve the equation for $x$ .

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Step 3.14.1

Remove parentheses.

$x^{2} = 7$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.14.2

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

$x = \pm \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.14.3

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

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Step 3.14.3.1

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

$x = \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.14.3.2

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

$x = - \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.14.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x = \sqrt{7}, - \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x = \sqrt{7}, - \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 3.15

The solution to $\frac{x^{4}}{16} - \frac{11 x^{2}}{8} + \frac{361}{16} = 16$ is $x = \sqrt{15}, - \sqrt{15}, \sqrt{7}, - \sqrt{7}$ .

$x = \sqrt{15}, - \sqrt{15}, \sqrt{7}, - \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

$x = \sqrt{15}, - \sqrt{15}, \sqrt{7}, - \sqrt{7}$

$y = - \frac{19}{4} + \frac{x^{2}}{4}$

Step 4

Replace all occurrences of $x$ with $\sqrt{15}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $\sqrt{15}$ .

$y = - \frac{19}{4} + \frac{{(\sqrt{15})}^{2}}{4}$

$x = \sqrt{15}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{19}{4} + \frac{{(\sqrt{15})}^{2}}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(\sqrt{15})}^{2}}{4}$

$x = \sqrt{15}$

Step 4.2.1.2

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(15^{\frac{1}{2}})}^{2}}{4}$

$x = \sqrt{15}$

Step 4.2.1.2.2

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

$y = \frac{- 19 + 15^{\frac{1}{2} \cdot 2}}{4}$

$x = \sqrt{15}$

Step 4.2.1.2.3

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

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = \sqrt{15}$

Step 4.2.1.2.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.4.1

Cancel the common factor.

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = \sqrt{15}$

Step 4.2.1.2.4.2

Rewrite the expression.

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

Step 4.2.1.2.5

Evaluate the exponent.

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Add $- 19$ and $15$ .

$y = \frac{- 4}{4}$

$x = \sqrt{15}$

Step 4.2.1.3.2

Divide $- 4$ by $4$ .

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

Step 5

Replace all occurrences of $x$ with $- \sqrt{15}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $- \sqrt{15}$ .

$y = - \frac{19}{4} + \frac{{(- \sqrt{15})}^{2}}{4}$

$x = - \sqrt{15}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{19}{4} + \frac{{(- \sqrt{15})}^{2}}{4}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(- \sqrt{15})}^{2}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

Apply the product rule to $- \sqrt{15}$ .

$y = \frac{- 19 + {(- 1)}^{2} {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.2

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

$y = \frac{- 19 + 1 {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.3

Multiply ${\sqrt{15}}^{2}$ by $1$ .

$y = \frac{- 19 + {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 5.2.1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(15^{\frac{1}{2}})}^{2}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.4.2

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

$y = \frac{- 19 + 15^{\frac{1}{2} \cdot 2}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.4.3

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

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.4.4

Cancel the common factor of $2$ .

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Step 5.2.1.2.4.4.1

Cancel the common factor.

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.4.4.2

Rewrite the expression.

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

Step 5.2.1.2.4.5

Evaluate the exponent.

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Add $- 19$ and $15$ .

$y = \frac{- 4}{4}$

$x = - \sqrt{15}$

Step 5.2.1.3.2

Divide $- 4$ by $4$ .

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

Step 6

Replace all occurrences of $x$ with $\sqrt{15}$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $\sqrt{15}$ .

$y = - \frac{19}{4} + \frac{{(\sqrt{15})}^{2}}{4}$

$x = \sqrt{15}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{19}{4} + \frac{{(\sqrt{15})}^{2}}{4}$ .

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Step 6.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(\sqrt{15})}^{2}}{4}$

$x = \sqrt{15}$

Step 6.2.1.2

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 6.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(15^{\frac{1}{2}})}^{2}}{4}$

$x = \sqrt{15}$

Step 6.2.1.2.2

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

$y = \frac{- 19 + 15^{\frac{1}{2} \cdot 2}}{4}$

$x = \sqrt{15}$

Step 6.2.1.2.3

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

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = \sqrt{15}$

Step 6.2.1.2.4

Cancel the common factor of $2$ .

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Step 6.2.1.2.4.1

Cancel the common factor.

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = \sqrt{15}$

Step 6.2.1.2.4.2

Rewrite the expression.

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

Step 6.2.1.2.5

Evaluate the exponent.

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

Step 6.2.1.3

Simplify the expression.

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Step 6.2.1.3.1

Add $- 19$ and $15$ .

$y = \frac{- 4}{4}$

$x = \sqrt{15}$

Step 6.2.1.3.2

Divide $- 4$ by $4$ .

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

Step 7

Replace all occurrences of $x$ with $- \sqrt{15}$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $- \sqrt{15}$ .

$y = - \frac{19}{4} + \frac{{(- \sqrt{15})}^{2}}{4}$

$x = - \sqrt{15}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{19}{4} + \frac{{(- \sqrt{15})}^{2}}{4}$ .

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Step 7.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(- \sqrt{15})}^{2}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2

Simplify each term.

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Step 7.2.1.2.1

Apply the product rule to $- \sqrt{15}$ .

$y = \frac{- 19 + {(- 1)}^{2} {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.2

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

$y = \frac{- 19 + 1 {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.3

Multiply ${\sqrt{15}}^{2}$ by $1$ .

$y = \frac{- 19 + {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 7.2.1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(15^{\frac{1}{2}})}^{2}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.4.2

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

$y = \frac{- 19 + 15^{\frac{1}{2} \cdot 2}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.4.3

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

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.4.4

Cancel the common factor of $2$ .

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Step 7.2.1.2.4.4.1

Cancel the common factor.

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.4.4.2

Rewrite the expression.

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

Step 7.2.1.2.4.5

Evaluate the exponent.

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

Step 7.2.1.3

Simplify the expression.

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Step 7.2.1.3.1

Add $- 19$ and $15$ .

$y = \frac{- 4}{4}$

$x = - \sqrt{15}$

Step 7.2.1.3.2

Divide $- 4$ by $4$ .

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

Step 8

Replace all occurrences of $x$ with $\sqrt{7}$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $\sqrt{7}$ .

$y = - \frac{19}{4} + \frac{{(\sqrt{7})}^{2}}{4}$

$x = \sqrt{7}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $- \frac{19}{4} + \frac{{(\sqrt{7})}^{2}}{4}$ .

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Step 8.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(\sqrt{7})}^{2}}{4}$

$x = \sqrt{7}$

Step 8.2.1.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 8.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(7^{\frac{1}{2}})}^{2}}{4}$

$x = \sqrt{7}$

Step 8.2.1.2.2

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

$y = \frac{- 19 + 7^{\frac{1}{2} \cdot 2}}{4}$

$x = \sqrt{7}$

Step 8.2.1.2.3

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

$y = \frac{- 19 + 7^{\frac{2}{2}}}{4}$

$x = \sqrt{7}$

Step 8.2.1.2.4

Cancel the common factor of $2$ .

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Step 8.2.1.2.4.1

Cancel the common factor.

$y = \frac{- 19 + 7^{\frac{2}{2}}}{4}$

$x = \sqrt{7}$

Step 8.2.1.2.4.2

Rewrite the expression.

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

Step 8.2.1.2.5

Evaluate the exponent.

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

Step 8.2.1.3

Simplify the expression.

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Step 8.2.1.3.1

Add $- 19$ and $7$ .

$y = \frac{- 12}{4}$

$x = \sqrt{7}$

Step 8.2.1.3.2

Divide $- 12$ by $4$ .

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

Step 9

Replace all occurrences of $x$ with $\sqrt{15}$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $\sqrt{15}$ .

$y = - \frac{19}{4} + \frac{{(\sqrt{15})}^{2}}{4}$

$x = \sqrt{15}$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $- \frac{19}{4} + \frac{{(\sqrt{15})}^{2}}{4}$ .

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Step 9.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(\sqrt{15})}^{2}}{4}$

$x = \sqrt{15}$

Step 9.2.1.2

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 9.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(15^{\frac{1}{2}})}^{2}}{4}$

$x = \sqrt{15}$

Step 9.2.1.2.2

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

$y = \frac{- 19 + 15^{\frac{1}{2} \cdot 2}}{4}$

$x = \sqrt{15}$

Step 9.2.1.2.3

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

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = \sqrt{15}$

Step 9.2.1.2.4

Cancel the common factor of $2$ .

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Step 9.2.1.2.4.1

Cancel the common factor.

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = \sqrt{15}$

Step 9.2.1.2.4.2

Rewrite the expression.

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

Step 9.2.1.2.5

Evaluate the exponent.

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = \sqrt{15}$

Step 9.2.1.3

Simplify the expression.

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Step 9.2.1.3.1

Add $- 19$ and $15$ .

$y = \frac{- 4}{4}$

$x = \sqrt{15}$

Step 9.2.1.3.2

Divide $- 4$ by $4$ .

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

$y = - 1$

$x = \sqrt{15}$

Step 10

Replace all occurrences of $x$ with $- \sqrt{15}$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $- \sqrt{15}$ .

$y = - \frac{19}{4} + \frac{{(- \sqrt{15})}^{2}}{4}$

$x = - \sqrt{15}$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $- \frac{19}{4} + \frac{{(- \sqrt{15})}^{2}}{4}$ .

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Step 10.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(- \sqrt{15})}^{2}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2

Simplify each term.

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Step 10.2.1.2.1

Apply the product rule to $- \sqrt{15}$ .

$y = \frac{- 19 + {(- 1)}^{2} {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.2

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

$y = \frac{- 19 + 1 {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.3

Multiply ${\sqrt{15}}^{2}$ by $1$ .

$y = \frac{- 19 + {\sqrt{15}}^{2}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 10.2.1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(15^{\frac{1}{2}})}^{2}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.4.2

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

$y = \frac{- 19 + 15^{\frac{1}{2} \cdot 2}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.4.3

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

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.4.4

Cancel the common factor of $2$ .

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Step 10.2.1.2.4.4.1

Cancel the common factor.

$y = \frac{- 19 + 15^{\frac{2}{2}}}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.4.4.2

Rewrite the expression.

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

Step 10.2.1.2.4.5

Evaluate the exponent.

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

$y = \frac{- 19 + 15}{4}$

$x = - \sqrt{15}$

Step 10.2.1.3

Simplify the expression.

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Step 10.2.1.3.1

Add $- 19$ and $15$ .

$y = \frac{- 4}{4}$

$x = - \sqrt{15}$

Step 10.2.1.3.2

Divide $- 4$ by $4$ .

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

$y = - 1$

$x = - \sqrt{15}$

Step 11

Replace all occurrences of $x$ with $\sqrt{7}$ in each equation.

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Step 11.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $\sqrt{7}$ .

$y = - \frac{19}{4} + \frac{{(\sqrt{7})}^{2}}{4}$

$x = \sqrt{7}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $- \frac{19}{4} + \frac{{(\sqrt{7})}^{2}}{4}$ .

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Step 11.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(\sqrt{7})}^{2}}{4}$

$x = \sqrt{7}$

Step 11.2.1.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 11.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(7^{\frac{1}{2}})}^{2}}{4}$

$x = \sqrt{7}$

Step 11.2.1.2.2

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

$y = \frac{- 19 + 7^{\frac{1}{2} \cdot 2}}{4}$

$x = \sqrt{7}$

Step 11.2.1.2.3

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

$y = \frac{- 19 + 7^{\frac{2}{2}}}{4}$

$x = \sqrt{7}$

Step 11.2.1.2.4

Cancel the common factor of $2$ .

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Step 11.2.1.2.4.1

Cancel the common factor.

$y = \frac{- 19 + 7^{\frac{2}{2}}}{4}$

$x = \sqrt{7}$

Step 11.2.1.2.4.2

Rewrite the expression.

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

Step 11.2.1.2.5

Evaluate the exponent.

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = \sqrt{7}$

Step 11.2.1.3

Simplify the expression.

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Step 11.2.1.3.1

Add $- 19$ and $7$ .

$y = \frac{- 12}{4}$

$x = \sqrt{7}$

Step 11.2.1.3.2

Divide $- 12$ by $4$ .

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

$y = - 3$

$x = \sqrt{7}$

Step 12

Replace all occurrences of $x$ with $- \sqrt{7}$ in each equation.

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Step 12.1

Replace all occurrences of $x$ in $y = - \frac{19}{4} + \frac{x^{2}}{4}$ with $- \sqrt{7}$ .

$y = - \frac{19}{4} + \frac{{(- \sqrt{7})}^{2}}{4}$

$x = - \sqrt{7}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $- \frac{19}{4} + \frac{{(- \sqrt{7})}^{2}}{4}$ .

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Step 12.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 19 + {(- \sqrt{7})}^{2}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2

Simplify each term.

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Step 12.2.1.2.1

Apply the product rule to $- \sqrt{7}$ .

$y = \frac{- 19 + {(- 1)}^{2} {\sqrt{7}}^{2}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.2

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

$y = \frac{- 19 + 1 {\sqrt{7}}^{2}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.3

Multiply ${\sqrt{7}}^{2}$ by $1$ .

$y = \frac{- 19 + {\sqrt{7}}^{2}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.4

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 12.2.1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \frac{- 19 + {(7^{\frac{1}{2}})}^{2}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.4.2

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

$y = \frac{- 19 + 7^{\frac{1}{2} \cdot 2}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.4.3

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

$y = \frac{- 19 + 7^{\frac{2}{2}}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.4.4

Cancel the common factor of $2$ .

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Step 12.2.1.2.4.4.1

Cancel the common factor.

$y = \frac{- 19 + 7^{\frac{2}{2}}}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.4.4.2

Rewrite the expression.

$y = \frac{- 19 + 7}{4}$

$x = - \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = - \sqrt{7}$

Step 12.2.1.2.4.5

Evaluate the exponent.

$y = \frac{- 19 + 7}{4}$

$x = - \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = - \sqrt{7}$

$y = \frac{- 19 + 7}{4}$

$x = - \sqrt{7}$

Step 12.2.1.3

Simplify the expression.

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Step 12.2.1.3.1

Add $- 19$ and $7$ .

$y = \frac{- 12}{4}$

$x = - \sqrt{7}$

Step 12.2.1.3.2

Divide $- 12$ by $4$ .

$y = - 3$

$x = - \sqrt{7}$

$y = - 3$

$x = - \sqrt{7}$

$y = - 3$

$x = - \sqrt{7}$

$y = - 3$

$x = - \sqrt{7}$

$y = - 3$

$x = - \sqrt{7}$

Step 13

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

$(\sqrt{15}, - 1)$

$(- \sqrt{15}, - 1)$

$(\sqrt{7}, - 3)$

$(- \sqrt{7}, - 3)$

Step 14

The result can be shown in multiple forms.

Point Form:

$(\sqrt{15}, - 1), (- \sqrt{15}, - 1), (\sqrt{7}, - 3), (- \sqrt{7}, - 3)$

Equation Form:

$x = \sqrt{15}, y = - 1$

$x = - \sqrt{15}, y = - 1$

$x = \sqrt{7}, y = - 3$

$x = - \sqrt{7}, y = - 3$

Step 15