Precalculus Examples

$3 y^{2} - x - 2 y + 1 = 0$

Step 1

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

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Step 1.1

Add $x$ to both sides of the equation.

$3 y^{2} - 2 y + 1 = x$

Step 1.2

Subtract $1$ from both sides of the equation.

$3 y^{2} - 2 y = x - 1$

Step 2

Complete the square.

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Step 2.1

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

$a = 3$

$b = - 2$

$c = 0 = x - 1$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = x - 1$

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor $2$ out of $2 \cdot 3$ .

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

Step 2.3.2.2

Move the negative in front of the fraction.

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

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

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

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 $2$ .

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

Step 2.4.2.1.2

Multiply $4$ by $3$ .

$e = 0 - \frac{4}{12}$

Step 2.4.2.1.3

Cancel the common factor of $4$ and $12$ .

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Step 2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = 0 - \frac{4 (1)}{12}$

Step 2.4.2.1.3.2

Cancel the common factors.

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Step 2.4.2.1.3.2.1

Factor $4$ out of $12$ .

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

Step 2.4.2.1.3.2.2

Cancel the common factor.

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

Step 2.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{1}{3}$

Step 2.4.2.2

Subtract $\frac{1}{3}$ from $0$ .

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

Step 2.5

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

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

Step 3

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

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Step 3.1

Add $\frac{1}{3}$ to both sides of the equation.

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

Step 3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.3

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $3$ .

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

Step 3.5.2

Add $- 3$ and $1$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Divide each term in $3 {(y - \frac{1}{3})}^{2} = x - \frac{2}{3}$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 {(y - \frac{1}{3})}^{2} = x - \frac{2}{3}$ by $3$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide ${(y - \frac{1}{3})}^{2}$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

Multiply $3$ by $3$ .

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

Step 5

Factor.

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Step 5.1

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1

Multiply $\frac{x}{3}$ by $\frac{3}{3}$ .

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

Step 5.2.2

Multiply $3$ by $3$ .

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 5.4

Move $3$ to the left of $x$ .

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

Step 5.5

Reorder terms.

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