Precalculus Examples

$4 x^{2} - 20 x + 24 y + 61 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

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

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Step 1.1.1.1

Subtract $4 x^{2}$ from both sides of the equation.

$- 20 x + 24 y + 61 = - 4 x^{2}$

Step 1.1.1.2

Add $20 x$ to both sides of the equation.

$24 y + 61 = - 4 x^{2} + 20 x$

Step 1.1.1.3

Subtract $61$ from both sides of the equation.

$24 y = - 4 x^{2} + 20 x - 61$

Step 1.1.2

Divide each term in $24 y = - 4 x^{2} + 20 x - 61$ by $24$ and simplify.

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Step 1.1.2.1

Divide each term in $24 y = - 4 x^{2} + 20 x - 61$ by $24$ .

$\frac{24 y}{24} = \frac{- 4 x^{2}}{24} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $24$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{24 y}{24} = \frac{- 4 x^{2}}{24} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 x^{2}}{24} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Cancel the common factor of $- 4$ and $24$ .

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Step 1.1.2.3.1.1.1

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

$y = \frac{4 (- x^{2})}{24} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.3.1.1.2

Cancel the common factors.

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Step 1.1.2.3.1.1.2.1

Factor $4$ out of $24$ .

$y = \frac{4 (- x^{2})}{4 (6)} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{4 (- x^{2})}{4 \cdot 6} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{- x^{2}}{6} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{6} + \frac{20 x}{24} + \frac{- 61}{24}$

Step 1.1.2.3.1.3

Cancel the common factor of $20$ and $24$ .

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Step 1.1.2.3.1.3.1

Factor $4$ out of $20 x$ .

$y = - \frac{x^{2}}{6} + \frac{4 (5 x)}{24} + \frac{- 61}{24}$

Step 1.1.2.3.1.3.2

Cancel the common factors.

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Step 1.1.2.3.1.3.2.1

Factor $4$ out of $24$ .

$y = - \frac{x^{2}}{6} + \frac{4 (5 x)}{4 (6)} + \frac{- 61}{24}$

Step 1.1.2.3.1.3.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{6} + \frac{4 (5 x)}{4 \cdot 6} + \frac{- 61}{24}$

Step 1.1.2.3.1.3.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{6} + \frac{5 x}{6} + \frac{- 61}{24}$

Step 1.1.2.3.1.4

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{6} + \frac{5 x}{6} - \frac{61}{24}$

Step 1.2

Complete the square for $- \frac{x^{2}}{6} + \frac{5 x}{6} - \frac{61}{24}$ .

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Step 1.2.1

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

$a = - \frac{1}{6}$

$b = \frac{5}{6}$

$c = - \frac{61}{24}$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{\frac{5}{6}}{2 (- \frac{1}{6})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{6} \cdot \frac{1}{2 (- \frac{1}{6})}$

Step 1.2.3.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 1.2.3.2.2.1

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

$d = \frac{5}{6} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{6})}$

Step 1.2.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{5}{6} (- \frac{1}{2 (\frac{1}{6})})$

Step 1.2.3.2.3

Combine $2$ and $\frac{1}{6}$ .

$d = \frac{5}{6} (- \frac{1}{\frac{2}{6}})$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{5}{6} (- \frac{1}{\frac{2 (1)}{6}})$

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $6$ .

$d = \frac{5}{6} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 3}})$

Step 1.2.3.2.4.2.2

Cancel the common factor.

$d = \frac{5}{6} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 3}})$

Step 1.2.3.2.4.2.3

Rewrite the expression.

$d = \frac{5}{6} (- \frac{1}{\frac{1}{3}})$

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{6} (- (1 \cdot 3))$

Step 1.2.3.2.6

Cancel the common factor of $3$ .

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Step 1.2.3.2.6.1

Factor $3$ out of $6$ .

$d = \frac{5}{3 (2)} (- (1 \cdot 3))$

Step 1.2.3.2.6.2

Factor $3$ out of $- (1 \cdot 3)$ .

$d = \frac{5}{3 (2)} (3 (- 1 \cdot (1)))$

Step 1.2.3.2.6.3

Cancel the common factor.

$d = \frac{5}{3 \cdot 2} (3 (- 1 \cdot (1)))$

Step 1.2.3.2.6.4

Rewrite the expression.

$d = \frac{5}{2} (- 1 \cdot (1))$

Step 1.2.3.2.7

Multiply $- 1$ by $1$ .

$d = \frac{5}{2} \cdot - 1$

Step 1.2.3.2.8

Combine $\frac{5}{2}$ and $- 1$ .

$d = \frac{5 \cdot - 1}{2}$

Step 1.2.3.2.9

Multiply $5$ by $- 1$ .

$d = \frac{- 5}{2}$

Step 1.2.3.2.10

Move the negative in front of the fraction.

$d = - \frac{5}{2}$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - \frac{61}{24} - \frac{{(\frac{5}{6})}^{2}}{4 (- \frac{1}{6})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{5}{6}$ .

$e = - \frac{61}{24} - \frac{\frac{5^{2}}{6^{2}}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.2

Raise $5$ to the power of $2$ .

$e = - \frac{61}{24} - \frac{\frac{25}{6^{2}}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.3

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

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{- 4 (\frac{1}{6})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{6}$ .

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{\frac{- 4}{6}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

Cancel the common factor of $- 4$ and $6$ .

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Step 1.2.4.2.1.3.1.1

Factor $2$ out of $- 4$ .

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{\frac{2 (- 2)}{6}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $2$ out of $6$ .

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{\frac{- 2}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{61}{24} - \frac{\frac{25}{36}}{- \frac{2}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{61}{24} - (\frac{25}{36} (- \frac{3}{2}))$

Step 1.2.4.2.1.5

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.5.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$e = - \frac{61}{24} - (\frac{25}{36} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.2

Factor $3$ out of $36$ .

$e = - \frac{61}{24} - (\frac{25}{3 (12)} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.3

Factor $3$ out of $- 3$ .

$e = - \frac{61}{24} - (\frac{25}{3 \cdot 12} \cdot \frac{3 \cdot - 1}{2})$

Step 1.2.4.2.1.5.4

Cancel the common factor.

$e = - \frac{61}{24} - (\frac{25}{3 \cdot 12} \cdot \frac{3 \cdot - 1}{2})$

Step 1.2.4.2.1.5.5

Rewrite the expression.

$e = - \frac{61}{24} - (\frac{25}{12} \cdot \frac{- 1}{2})$

Step 1.2.4.2.1.6

Multiply $\frac{25}{12}$ by $\frac{- 1}{2}$ .

$e = - \frac{61}{24} - \frac{25 \cdot - 1}{12 \cdot 2}$

Step 1.2.4.2.1.7

Multiply $25$ by $- 1$ .

$e = - \frac{61}{24} - \frac{- 25}{12 \cdot 2}$

Step 1.2.4.2.1.8

Multiply $12$ by $2$ .

$e = - \frac{61}{24} - \frac{- 25}{24}$

Step 1.2.4.2.1.9

Move the negative in front of the fraction.

$e = - \frac{61}{24} - - \frac{25}{24}$

Step 1.2.4.2.1.10

Multiply $- - \frac{25}{24}$ .

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Step 1.2.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{61}{24} + 1 (\frac{25}{24})$

Step 1.2.4.2.1.10.2

Multiply $\frac{25}{24}$ by $1$ .

$e = - \frac{61}{24} + \frac{25}{24}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 61 + 25}{24}$

Step 1.2.4.2.3

Add $- 61$ and $25$ .

$e = \frac{- 36}{24}$

Step 1.2.4.2.4

Cancel the common factor of $- 36$ and $24$ .

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Step 1.2.4.2.4.1

Factor $12$ out of $- 36$ .

$e = \frac{12 (- 3)}{24}$

Step 1.2.4.2.4.2

Cancel the common factors.

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Step 1.2.4.2.4.2.1

Factor $12$ out of $24$ .

$e = \frac{12 \cdot - 3}{12 \cdot 2}$

Step 1.2.4.2.4.2.2

Cancel the common factor.

$e = \frac{12 \cdot - 3}{12 \cdot 2}$

Step 1.2.4.2.4.2.3

Rewrite the expression.

$e = \frac{- 3}{2}$

Step 1.2.4.2.5

Move the negative in front of the fraction.

$e = - \frac{3}{2}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{6} {(x - \frac{5}{2})}^{2} - \frac{3}{2}$ .

$- \frac{1}{6} {(x - \frac{5}{2})}^{2} - \frac{3}{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{6} \cdot {(x - \frac{5}{2})}^{2} - \frac{3}{2}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{6}$

$h = \frac{5}{2}$

$k = - \frac{3}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{5}{2}, - \frac{3}{2})$

Step 4