Precalculus Examples

$x^{2} - 3 x + y + y^{2} = 81$

Step 1

Complete the square for $x^{2} - 3 x$ .

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Step 1.1

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

$a = 1$

$b = - 3$

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Multiply $2$ by $1$ .

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

Step 1.3.2.2

Move the negative in front of the fraction.

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

Step 1.4

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

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Step 1.4.1

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

$e = 0 - \frac{{(- 3)}^{2}}{4 \cdot 1}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Raise $- 3$ to the power of $2$ .

$e = 0 - \frac{9}{4 \cdot 1}$

Step 1.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.4.2.2

Subtract $\frac{9}{4}$ from $0$ .

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

Step 1.5

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

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

Step 2

Substitute ${(x - \frac{3}{2})}^{2} - \frac{9}{4}$ for $x^{2} - 3 x$ in the equation $x^{2} - 3 x + y + y^{2} = 81$ .

${(x - \frac{3}{2})}^{2} - \frac{9}{4} + y + y^{2} = 81$

Step 3

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

${(x - \frac{3}{2})}^{2} + y + y^{2} = 81 + \frac{9}{4}$

Step 4

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

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Step 4.1

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

$y^{2} + y$

Step 4.2

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

$a = 1$

$b = 1$

$c = 0$

Step 4.3

Consider the vertex form of a parabola.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factor of $1$ .

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Rewrite the expression.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

One to any power is one.

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

Step 4.5.2.1.2

Multiply $4$ by $1$ .

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

Step 4.5.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 4.6

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

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

Step 5

Substitute ${(y + \frac{1}{2})}^{2} - \frac{1}{4}$ for $y + y^{2}$ in the equation $x^{2} - 3 x + y + y^{2} = 81$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} - \frac{1}{4} = 81 + \frac{9}{4}$

Step 6

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{9}{4} + \frac{1}{4}$

Step 7

Simplify $81 + \frac{9}{4} + \frac{1}{4}$ .

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Step 7.1

Combine the numerators over the common denominator.

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{9 + 1}{4}$

Step 7.2

Add $9$ and $1$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{10}{4}$

Step 7.3

Cancel the common factor of $10$ and $4$ .

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Step 7.3.1

Factor $2$ out of $10$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{2 (5)}{4}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $4$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{2 \cdot 5}{2 \cdot 2}$

Step 7.3.2.2

Cancel the common factor.

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{2 \cdot 5}{2 \cdot 2}$

Step 7.3.2.3

Rewrite the expression.

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 + \frac{5}{2}$

Step 7.4

To write $81$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = 81 \cdot \frac{2}{2} + \frac{5}{2}$

Step 7.5

Combine $81$ and $\frac{2}{2}$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{81 \cdot 2}{2} + \frac{5}{2}$

Step 7.6

Combine the numerators over the common denominator.

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{81 \cdot 2 + 5}{2}$

Step 7.7

Simplify the numerator.

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Step 7.7.1

Multiply $81$ by $2$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{162 + 5}{2}$

Step 7.7.2

Add $162$ and $5$ .

${(x - \frac{3}{2})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{167}{2}$