Precalculus Examples

$x^{2} + y^{2} + \frac{3}{4} x + \frac{5}{3} y - \frac{1}{2} = 0$

Step 1

Add $\frac{1}{2}$ to both sides of the equation.

$x^{2} + y^{2} + \frac{3}{4} x + \frac{5}{3} y = \frac{1}{2}$

Step 2

Complete the square for $x^{2} + \frac{3}{4} x$ .

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Step 2.1

Combine $\frac{3}{4}$ and $x$ .

$x^{2} + \frac{3 x}{4}$

Step 2.2

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

$a = 1$

$b = \frac{3}{4}$

$c = 0$

Step 2.3

Consider the vertex form of a parabola.

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

Step 2.4

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

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Step 2.4.1

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

$d = \frac{\frac{3}{4}}{2 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} \cdot \frac{1}{2 \cdot 1}$

Step 2.4.2.2

Cancel the common factor of $1$ .

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Step 2.4.2.2.1

Cancel the common factor.

$d = \frac{3}{4} \cdot \frac{1}{2 \cdot 1}$

Step 2.4.2.2.2

Rewrite the expression.

$d = \frac{3}{4} \cdot \frac{1}{2}$

Step 2.4.2.3

Multiply $\frac{3}{4} \cdot \frac{1}{2}$ .

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Step 2.4.2.3.1

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

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

Step 2.4.2.3.2

Multiply $4$ by $2$ .

$d = \frac{3}{8}$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Simplify the numerator.

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Step 2.5.2.1.1.1

Apply the product rule to $\frac{3}{4}$ .

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

Step 2.5.2.1.1.2

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

$e = 0 - \frac{\frac{9}{4^{2}}}{4 \cdot 1}$

Step 2.5.2.1.1.3

Raise $4$ to the power of $2$ .

$e = 0 - \frac{\frac{9}{16}}{4 \cdot 1}$

Step 2.5.2.1.2

Multiply $4$ by $1$ .

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

Step 2.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{9}{16} \cdot \frac{1}{4})$

Step 2.5.2.1.4

Multiply $\frac{9}{16} \cdot \frac{1}{4}$ .

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Step 2.5.2.1.4.1

Multiply $\frac{9}{16}$ by $\frac{1}{4}$ .

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

Step 2.5.2.1.4.2

Multiply $16$ by $4$ .

$e = 0 - \frac{9}{64}$

Step 2.5.2.2

Subtract $\frac{9}{64}$ from $0$ .

$e = - \frac{9}{64}$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{3}{8})}^{2} - \frac{9}{64}$ .

${(x + \frac{3}{8})}^{2} - \frac{9}{64}$

Step 3

Substitute ${(x + \frac{3}{8})}^{2} - \frac{9}{64}$ for $x^{2} + \frac{3}{4} x$ in the equation $x^{2} + y^{2} + \frac{3}{4} x + \frac{5}{3} y = \frac{1}{2}$ .

${(x + \frac{3}{8})}^{2} - \frac{9}{64} + y^{2} + \frac{5}{3} y = \frac{1}{2}$

Step 4

Move $- \frac{9}{64}$ to the right side of the equation by adding $\frac{9}{64}$ to both sides.

${(x + \frac{3}{8})}^{2} + y^{2} + \frac{5}{3} y = \frac{1}{2} + \frac{9}{64}$

Step 5

Complete the square for $y^{2} + \frac{5}{3} y$ .

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Step 5.1

Combine $\frac{5}{3}$ and $y$ .

$y^{2} + \frac{5 y}{3}$

Step 5.2

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

$a = 1$

$b = \frac{5}{3}$

$c = 0$

Step 5.3

Consider the vertex form of a parabola.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4.2.2

Cancel the common factor of $1$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Rewrite the expression.

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

Step 5.4.2.3

Multiply $\frac{5}{3} \cdot \frac{1}{2}$ .

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Step 5.4.2.3.1

Multiply $\frac{5}{3}$ by $\frac{1}{2}$ .

$d = \frac{5}{3 \cdot 2}$

Step 5.4.2.3.2

Multiply $3$ by $2$ .

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

Step 5.5

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

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Step 5.5.1

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

$e = 0 - \frac{{(\frac{5}{3})}^{2}}{4 \cdot 1}$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Simplify each term.

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Step 5.5.2.1.1

Simplify the numerator.

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Step 5.5.2.1.1.1

Apply the product rule to $\frac{5}{3}$ .

$e = 0 - \frac{\frac{5^{2}}{3^{2}}}{4 \cdot 1}$

Step 5.5.2.1.1.2

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

$e = 0 - \frac{\frac{25}{3^{2}}}{4 \cdot 1}$

Step 5.5.2.1.1.3

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

$e = 0 - \frac{\frac{25}{9}}{4 \cdot 1}$

Step 5.5.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{25}{9}}{4}$

Step 5.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{25}{9} \cdot \frac{1}{4})$

Step 5.5.2.1.4

Multiply $\frac{25}{9} \cdot \frac{1}{4}$ .

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Step 5.5.2.1.4.1

Multiply $\frac{25}{9}$ by $\frac{1}{4}$ .

$e = 0 - \frac{25}{9 \cdot 4}$

Step 5.5.2.1.4.2

Multiply $9$ by $4$ .

$e = 0 - \frac{25}{36}$

Step 5.5.2.2

Subtract $\frac{25}{36}$ from $0$ .

$e = - \frac{25}{36}$

Step 5.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + \frac{5}{6})}^{2} - \frac{25}{36}$ .

${(y + \frac{5}{6})}^{2} - \frac{25}{36}$

Step 6

Substitute ${(y + \frac{5}{6})}^{2} - \frac{25}{36}$ for $y^{2} + \frac{5}{3} y$ in the equation $x^{2} + y^{2} + \frac{3}{4} x + \frac{5}{3} y = \frac{1}{2}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} - \frac{25}{36} = \frac{1}{2} + \frac{9}{64}$

Step 7

Move $- \frac{25}{36}$ to the right side of the equation by adding $\frac{25}{36}$ to both sides.

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{1}{2} + \frac{9}{64} + \frac{25}{36}$

Step 8

Simplify $\frac{1}{2} + \frac{9}{64} + \frac{25}{36}$ .

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Step 8.1

Find the common denominator.

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Step 8.1.1

Multiply $\frac{1}{2}$ by $\frac{288}{288}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{1}{2} \cdot \frac{288}{288} + \frac{9}{64} + \frac{25}{36}$

Step 8.1.2

Multiply $\frac{1}{2}$ by $\frac{288}{288}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{2 \cdot 288} + \frac{9}{64} + \frac{25}{36}$

Step 8.1.3

Multiply $\frac{9}{64}$ by $\frac{9}{9}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{2 \cdot 288} + \frac{9}{64} \cdot \frac{9}{9} + \frac{25}{36}$

Step 8.1.4

Multiply $\frac{9}{64}$ by $\frac{9}{9}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{2 \cdot 288} + \frac{9 \cdot 9}{64 \cdot 9} + \frac{25}{36}$

Step 8.1.5

Multiply $\frac{25}{36}$ by $\frac{16}{16}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{2 \cdot 288} + \frac{9 \cdot 9}{64 \cdot 9} + \frac{25}{36} \cdot \frac{16}{16}$

Step 8.1.6

Multiply $\frac{25}{36}$ by $\frac{16}{16}$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{2 \cdot 288} + \frac{9 \cdot 9}{64 \cdot 9} + \frac{25 \cdot 16}{36 \cdot 16}$

Step 8.1.7

Multiply $2$ by $288$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{576} + \frac{9 \cdot 9}{64 \cdot 9} + \frac{25 \cdot 16}{36 \cdot 16}$

Step 8.1.8

Reorder the factors of $64 \cdot 9$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{576} + \frac{9 \cdot 9}{9 \cdot 64} + \frac{25 \cdot 16}{36 \cdot 16}$

Step 8.1.9

Multiply $9$ by $64$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{576} + \frac{9 \cdot 9}{576} + \frac{25 \cdot 16}{36 \cdot 16}$

Step 8.1.10

Reorder the factors of $36 \cdot 16$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{576} + \frac{9 \cdot 9}{576} + \frac{25 \cdot 16}{16 \cdot 36}$

Step 8.1.11

Multiply $16$ by $36$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288}{576} + \frac{9 \cdot 9}{576} + \frac{25 \cdot 16}{576}$

Step 8.2

Combine the numerators over the common denominator.

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288 + 9 \cdot 9 + 25 \cdot 16}{576}$

Step 8.3

Simplify each term.

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Step 8.3.1

Multiply $9$ by $9$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288 + 81 + 25 \cdot 16}{576}$

Step 8.3.2

Multiply $25$ by $16$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{288 + 81 + 400}{576}$

Step 8.4

Simplify by adding numbers.

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Step 8.4.1

Add $288$ and $81$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{369 + 400}{576}$

Step 8.4.2

Add $369$ and $400$ .

${(x + \frac{3}{8})}^{2} + {(y + \frac{5}{6})}^{2} = \frac{769}{576}$

Step 9

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 10

Match the values in this circle to those of the standard form. The variable $r$ represents the radius of the circle, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from origin.

$r = \frac{\sqrt{769}}{24}$

$h = - \frac{3}{8}$

$k = - \frac{5}{6}$

Step 11

The center of the circle is found at $(h, k)$ .

Center: $(- \frac{3}{8}, - \frac{5}{6})$

Step 12

These values represent the important values for graphing and analyzing a circle.

Center: $(- \frac{3}{8}, - \frac{5}{6})$

Radius: $\frac{\sqrt{769}}{24}$

Step 13