Precalculus Examples

$x^{2} + 6 x - 4 y + 1 = 0$

Step 1

Isolate $y$ to the left side of the equation.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Subtract $x^{2}$ from both sides of the equation.

$6 x - 4 y + 1 = - x^{2}$

Step 1.1.2

Subtract $6 x$ from both sides of the equation.

$- 4 y + 1 = - x^{2} - 6 x$

Step 1.1.3

Subtract $1$ from both sides of the equation.

$- 4 y = - x^{2} - 6 x - 1$

Step 1.2

Divide each term in $- 4 y = - x^{2} - 6 x - 1$ by $- 4$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in $- 4 y = - x^{2} - 6 x - 1$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- x^{2}}{- 4} + \frac{- 6 x}{- 4} + \frac{- 1}{- 4}$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- x^{2}}{- 4} + \frac{- 6 x}{- 4} + \frac{- 1}{- 4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 4} + \frac{- 6 x}{- 4} + \frac{- 1}{- 4}$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{4} + \frac{- 6 x}{- 4} + \frac{- 1}{- 4}$

Step 1.2.3.1.2

Cancel the common factor of $- 6$ and $- 4$ .

Tap for more steps...

Step 1.2.3.1.2.1

Factor $- 2$ out of $- 6 x$ .

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

Step 1.2.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.1.2.2.1

Factor $- 2$ out of $- 4$ .

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

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

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

Step 2

Complete the square for $\frac{x^{2}}{4} + \frac{3 x}{2} + \frac{1}{4}$ .

Tap for more steps...

Step 2.1

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

$a = \frac{1}{4}$

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

$c = \frac{1}{4}$

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

Tap for more steps...

Step 2.3.1

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

$d = \frac{\frac{3}{2}}{2 (\frac{1}{4})}$

Step 2.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.2.2

Combine $2$ and $\frac{1}{4}$ .

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

Step 2.3.2.3

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 2.3.2.3.1

Factor $2$ out of $2$ .

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

Step 2.3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.3.2.1

Factor $2$ out of $4$ .

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

Step 2.3.2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3.2.3

Rewrite the expression.

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

Step 2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.2.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.5.1

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

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

Step 2.3.2.5.2

Cancel the common factor.

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

Step 2.3.2.5.3

Rewrite the expression.

$d = 3$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify each term.

Tap for more steps...

Step 2.4.2.1.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.1.1.1

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

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

Step 2.4.2.1.1.2

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

$e = \frac{1}{4} - \frac{\frac{9}{2^{2}}}{4 (\frac{1}{4})}$

Step 2.4.2.1.1.3

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

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

Step 2.4.2.1.2

Combine $4$ and $\frac{1}{4}$ .

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

Step 2.4.2.1.3

Divide $4$ by $4$ .

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

Step 2.4.2.1.4

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

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

Step 2.4.2.2

Combine the numerators over the common denominator.

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

Step 2.4.2.3

Subtract $9$ from $1$ .

$e = \frac{- 8}{4}$

Step 2.4.2.4

Divide $- 8$ by $4$ .

$e = - 2$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x + 3)}^{2} - 2$ .

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

Step 3

Set $y$ equal to the new right side.

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