Precalculus Examples

$4 x^{2} - 4 y^{2} - 4 x + 6 y + 17 = 0$

Step 1

Subtract $17$ from both sides of the equation.

$4 x^{2} - 4 y^{2} - 4 x + 6 y = - 17$

Step 2

Complete the square for $4 x^{2} - 4 x$ .

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Step 2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 4$

$b = - 4$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 \cdot 4}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot 4}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 2}{2 (4)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 4}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 2}{4}$

Step 2.3.2.2

Cancel the common factor of $- 2$ and $4$ .

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Step 2.3.2.2.1

Factor $2$ out of $- 2$ .

$d = \frac{2 (- 1)}{4}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 1}{2}$

Step 2.3.2.3

Move the negative in front of the fraction.

$d = - \frac{1}{2}$

Step 2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot 4}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 2.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot 4}$

Step 2.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 4}$

Step 2.4.2.1.1.3

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

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot 4}$

Step 2.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 0 - \frac{4^{2}}{4 \cdot 4}$

Step 2.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 4}$

Step 2.4.2.1.1.6

Cancel the common factors.

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Step 2.4.2.1.1.6.1

Factor $4$ out of $4 \cdot 4$ .

$e = 0 - \frac{4 \cdot 4}{4 (4)}$

Step 2.4.2.1.1.6.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 4}$

Step 2.4.2.1.1.6.3

Rewrite the expression.

$e = 0 - \frac{4}{4}$

Step 2.4.2.1.2

Cancel the common factor of $4$ .

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Step 2.4.2.1.2.1

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Step 2.4.2.1.2.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 2.4.2.1.3

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - \frac{1}{2})}^{2} - 1$ .

$4 {(x - \frac{1}{2})}^{2} - 1$

Step 3

Substitute $4 {(x - \frac{1}{2})}^{2} - 1$ for $4 x^{2} - 4 x$ in the equation $4 x^{2} - 4 y^{2} - 4 x + 6 y = - 17$ .

$4 {(x - \frac{1}{2})}^{2} - 1 - 4 y^{2} + 6 y = - 17$

Step 4

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

$4 {(x - \frac{1}{2})}^{2} - 4 y^{2} + 6 y = - 17 + 1$

Step 5

Complete the square for $- 4 y^{2} + 6 y$ .

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Step 5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 4$

$b = 6$

$c = 0$

Step 5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{6}{2 \cdot - 4}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $6$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 \cdot - 4}$

Step 5.3.2.1.2

Cancel the common factors.

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Step 5.3.2.1.2.1

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot 3}{2 (- 4)}$

Step 5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot - 4}$

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{3}{- 4}$

Step 5.3.2.2

Move the negative in front of the fraction.

$d = - \frac{3}{4}$

Step 5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{6^{2}}{4 \cdot - 4}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Raise $6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot - 4}$

Step 5.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{36}{- 16}$

Step 5.4.2.1.3

Cancel the common factor of $36$ and $- 16$ .

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Step 5.4.2.1.3.1

Factor $4$ out of $36$ .

$e = 0 - \frac{4 (9)}{- 16}$

Step 5.4.2.1.3.2

Cancel the common factors.

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Step 5.4.2.1.3.2.1

Factor $4$ out of $- 16$ .

$e = 0 - \frac{4 \cdot 9}{4 \cdot - 4}$

Step 5.4.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 9}{4 \cdot - 4}$

Step 5.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{9}{- 4}$

Step 5.4.2.1.4

Move the negative in front of the fraction.

$e = 0 - - \frac{9}{4}$

Step 5.4.2.1.5

Multiply $- - \frac{9}{4}$ .

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Step 5.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{9}{4})$

Step 5.4.2.1.5.2

Multiply $\frac{9}{4}$ by $1$ .

$e = 0 + \frac{9}{4}$

Step 5.4.2.2

Add $0$ and $\frac{9}{4}$ .

$e = \frac{9}{4}$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(y - \frac{3}{4})}^{2} + \frac{9}{4}$ .

$- 4 {(y - \frac{3}{4})}^{2} + \frac{9}{4}$

Step 6

Substitute $- 4 {(y - \frac{3}{4})}^{2} + \frac{9}{4}$ for $- 4 y^{2} + 6 y$ in the equation $4 x^{2} - 4 y^{2} - 4 x + 6 y = - 17$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} + \frac{9}{4} = - 17 + 1$

Step 7

Move $\frac{9}{4}$ to the right side of the equation by adding $\frac{9}{4}$ to both sides.

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = - 17 + 1 - \frac{9}{4}$

Step 8

Simplify $- 17 + 1 - \frac{9}{4}$ .

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Step 8.1

Find the common denominator.

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Step 8.1.1

Write $- 17$ as a fraction with denominator $1$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17}{1} + 1 - \frac{9}{4}$

Step 8.1.2

Multiply $\frac{- 17}{1}$ by $\frac{4}{4}$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17}{1} \cdot \frac{4}{4} + 1 - \frac{9}{4}$

Step 8.1.3

Multiply $\frac{- 17}{1}$ by $\frac{4}{4}$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17 \cdot 4}{4} + 1 - \frac{9}{4}$

Step 8.1.4

Write $1$ as a fraction with denominator $1$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17 \cdot 4}{4} + \frac{1}{1} - \frac{9}{4}$

Step 8.1.5

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

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17 \cdot 4}{4} + \frac{1}{1} \cdot \frac{4}{4} - \frac{9}{4}$

Step 8.1.6

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

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17 \cdot 4}{4} + \frac{4}{4} - \frac{9}{4}$

Step 8.2

Combine the numerators over the common denominator.

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 17 \cdot 4 + 4 - 9}{4}$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $- 17$ by $4$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 68 + 4 - 9}{4}$

Step 8.3.2

Add $- 68$ and $4$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 64 - 9}{4}$

Step 8.3.3

Subtract $9$ from $- 64$ .

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = \frac{- 73}{4}$

Step 8.3.4

Move the negative in front of the fraction.

$4 {(x - \frac{1}{2})}^{2} - 4 {(y - \frac{3}{4})}^{2} = - \frac{73}{4}$

Step 9

Flip the sign on each term of the equation so the term on the right side is positive.

$- 4 {(x - \frac{1}{2})}^{2} + 4 {(y - \frac{3}{4})}^{2} = \frac{73}{4}$

Step 10

Divide each term by $\frac{73}{4}$ to make the right side equal to one.

$- \frac{4 {(x - \frac{1}{2})}^{2}}{\frac{73}{4}} + \frac{4 {(y - \frac{3}{4})}^{2}}{\frac{73}{4}} = \frac{\frac{73}{4}}{\frac{73}{4}}$

Step 11

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(y - \frac{3}{4})}^{2}}{\frac{73}{16}} - \frac{{(x - \frac{1}{2})}^{2}}{\frac{73}{16}} = 1$