Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

$x^{2} + y^{2} - \frac{2}{3} x - 3 y = - 2$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{2}{3}$ .

$x^{2} - \frac{x \cdot 2}{3}$

Step 2.1.2

Move $2$ to the left of $x$ .

$x^{2} - \frac{2 x}{3}$

Step 2.2

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

$a = 1$

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

$c = 0$

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

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

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

Step 2.4.2.1.2

Cancel the common factor.

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

Step 2.4.2.1.3

Rewrite the expression.

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

Step 2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.2.3

Cancel the common factor of $2$ .

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Step 2.4.2.3.1

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

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

Step 2.4.2.3.2

Factor $2$ out of $- 2$ .

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

Step 2.4.2.3.3

Cancel the common factor.

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

Step 2.4.2.3.4

Rewrite the expression.

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

Step 2.4.2.4

Move the negative in front of the fraction.

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

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

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

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 $- \frac{2}{3}$ .

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

Step 2.5.2.1.1.2

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

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

Step 2.5.2.1.1.3

Apply the product rule to $\frac{2}{3}$ .

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

Step 2.5.2.1.1.4

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

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

Step 2.5.2.1.1.5

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

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

Step 2.5.2.1.1.6

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

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

Step 2.5.2.1.2

Multiply $4$ by $1$ .

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

Step 2.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.2.1.4

Cancel the common factor of $4$ .

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Step 2.5.2.1.4.1

Cancel the common factor.

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

Step 2.5.2.1.4.2

Rewrite the expression.

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

Step 2.5.2.2

Subtract $\frac{1}{9}$ from $0$ .

$e = - \frac{1}{9}$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{1}{3})}^{2} - \frac{1}{9}$ .

${(x - \frac{1}{3})}^{2} - \frac{1}{9}$

Step 3

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

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

Step 4

Move $- \frac{1}{9}$ to the right side of the equation by adding $\frac{1}{9}$ to both sides.

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

Step 5

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

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Step 5.1

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

$a = 1$

$b = - 3$

$c = 0$

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

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $2$ by $1$ .

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

Step 5.3.2.2

Move the negative in front of the fraction.

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

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

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

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

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

Step 5.4.2.1.2

Multiply $4$ by $1$ .

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

Step 5.4.2.2

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

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

Step 5.5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify $- 2 + \frac{1}{9} + \frac{9}{4}$ .

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Step 8.1

Find the common denominator.

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Step 8.1.1

Write $- 2$ as a fraction with denominator $1$ .

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

Step 8.1.2

Multiply $\frac{- 2}{1}$ by $\frac{36}{36}$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2}{1} \cdot \frac{36}{36} + \frac{1}{9} + \frac{9}{4}$

Step 8.1.3

Multiply $\frac{- 2}{1}$ by $\frac{36}{36}$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{1}{9} + \frac{9}{4}$

Step 8.1.4

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

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{1}{9} \cdot \frac{4}{4} + \frac{9}{4}$

Step 8.1.5

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

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{4}{9 \cdot 4} + \frac{9}{4}$

Step 8.1.6

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

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{4}{9 \cdot 4} + \frac{9}{4} \cdot \frac{9}{9}$

Step 8.1.7

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

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{4}{9 \cdot 4} + \frac{9 \cdot 9}{4 \cdot 9}$

Step 8.1.8

Reorder the factors of $9 \cdot 4$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{4}{4 \cdot 9} + \frac{9 \cdot 9}{4 \cdot 9}$

Step 8.1.9

Multiply $4$ by $9$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{4}{36} + \frac{9 \cdot 9}{4 \cdot 9}$

Step 8.1.10

Multiply $4$ by $9$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36}{36} + \frac{4}{36} + \frac{9 \cdot 9}{36}$

Step 8.2

Combine the numerators over the common denominator.

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 2 \cdot 36 + 4 + 9 \cdot 9}{36}$

Step 8.3

Simplify each term.

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Step 8.3.1

Multiply $- 2$ by $36$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 72 + 4 + 9 \cdot 9}{36}$

Step 8.3.2

Multiply $9$ by $9$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 72 + 4 + 81}{36}$

Step 8.4

Simplify by adding numbers.

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Step 8.4.1

Add $- 72$ and $4$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{- 68 + 81}{36}$

Step 8.4.2

Add $- 68$ and $81$ .

${(x - \frac{1}{3})}^{2} + {(y - \frac{3}{2})}^{2} = \frac{13}{36}$