Precalculus Examples

$y^{2} + 2 x + x^{2} = 24 y - 120$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $24 y$ from both sides of the equation.

$y^{2} + 2 x + x^{2} - 24 y = - 120$

Step 1.2

Move $2 x$ .

$y^{2} + x^{2} + 2 x - 24 y = - 120$

Step 1.3

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

$x^{2} + y^{2} + 2 x - 24 y = - 120$

Step 2

Complete the square for $x^{2} + 2 x$ .

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Step 2.1

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

$a = 1$

$b = 2$

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

$d = 1$

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 $e = c - \frac{b^{2}}{4 a}$ .

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

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

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

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Cancel the common factor of $4$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

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

Step 2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 2.4.2.1.4

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 ${(x + 1)}^{2} - 1$ .

${(x + 1)}^{2} - 1$

Step 3

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + y^{2} + 2 x - 24 y = - 120$ .

${(x + 1)}^{2} - 1 + y^{2} - 24 y = - 120$

Step 4

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

${(x + 1)}^{2} + y^{2} - 24 y = - 120 + 1$

Step 5

Complete the square for $y^{2} - 24 y$ .

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Step 5.1

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

$a = 1$

$b = - 24$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 24}{2 \cdot 1}$

Step 5.3.2

Cancel the common factor of $- 24$ and $2$ .

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Step 5.3.2.1

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot 1}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 12}{2 (1)}$

Step 5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot 1}$

Step 5.3.2.2.3

Rewrite the expression.

$d = \frac{- 12}{1}$

Step 5.3.2.2.4

Divide $- 12$ by $1$ .

$d = - 12$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot 1}$

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 $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{576}{4}$

Step 5.4.2.1.3

Divide $576$ by $4$ .

$e = 0 - 1 \cdot 144$

Step 5.4.2.1.4

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Step 5.4.2.2

Subtract $144$ from $0$ .

$e = - 144$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 12)}^{2} - 144$ .

${(y - 12)}^{2} - 144$

Step 6

Substitute ${(y - 12)}^{2} - 144$ for $y^{2} - 24 y$ in the equation $x^{2} + y^{2} + 2 x - 24 y = - 120$ .

${(x + 1)}^{2} + {(y - 12)}^{2} - 144 = - 120 + 1$

Step 7

Move $- 144$ to the right side of the equation by adding $144$ to both sides.

${(x + 1)}^{2} + {(y - 12)}^{2} = - 120 + 1 + 144$

Step 8

Simplify $- 120 + 1 + 144$ .

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Step 8.1

Add $- 120$ and $1$ .

${(x + 1)}^{2} + {(y - 12)}^{2} = - 119 + 144$

Step 8.2

Add $- 119$ and $144$ .

${(x + 1)}^{2} + {(y - 12)}^{2} = 25$