Find the Vertex x^2+6x-8y+1=0

$x^{2} + 6 x - 8 y + 1 = 0$

Step 1

Rewrite the equation in vertex form.

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

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

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

$6 x - 8 y + 1 = - x^{2}$

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

$- 8 y + 1 = - x^{2} - 6 x$

Subtract $1$ from both sides of the equation.

$- 8 y = - x^{2} - 6 x - 1$

Divide each term in $- 8 y = - x^{2} - 6 x - 1$ by $- 8$ and simplify.

Divide each term in $- 8 y = - x^{2} - 6 x - 1$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{- 6 x}{- 8} + \frac{- 1}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{- 6 x}{- 8} + \frac{- 1}{- 8}$

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 8} + \frac{- 6 x}{- 8} + \frac{- 1}{- 8}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{8} + \frac{- 6 x}{- 8} + \frac{- 1}{- 8}$

Cancel the common factor of $- 6$ and $- 8$ .

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

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

Cancel the common factors.

Factor $- 2$ out of $- 8$ .

$y = \frac{x^{2}}{8} + \frac{- 2 (3 x)}{- 2 (4)} + \frac{- 1}{- 8}$

Cancel the common factor.

$y = \frac{x^{2}}{8} + \frac{- 2 (3 x)}{- 2 \cdot 4} + \frac{- 1}{- 8}$

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{- 1}{- 8}$

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{1}{8}$

Complete the square for $\frac{x^{2}}{8} + \frac{3 x}{4} + \frac{1}{8}$ .

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

$a = \frac{1}{8}$

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

$c = \frac{1}{8}$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{\frac{3}{4}}{2 (\frac{1}{8})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} \cdot \frac{1}{2 (\frac{1}{8})}$

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2}{8}}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 (1)}{8}}$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $4$ .

Factor $4$ out of $1 \cdot 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = 3$

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

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

$e = \frac{1}{8} - \frac{{(\frac{3}{4})}^{2}}{4 (\frac{1}{8})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

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

$e = \frac{1}{8} - \frac{\frac{3^{2}}{4^{2}}}{4 (\frac{1}{8})}$

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

$e = \frac{1}{8} - \frac{\frac{9}{4^{2}}}{4 (\frac{1}{8})}$

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

$e = \frac{1}{8} - \frac{\frac{9}{16}}{4 (\frac{1}{8})}$

Combine $4$ and $\frac{1}{8}$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4}{8}}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Cancel the common factor.

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Rewrite the expression.

$e = \frac{1}{8} - \frac{\frac{9}{16}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{8} - (\frac{9}{16} \cdot 2)$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$e = \frac{1}{8} - (\frac{9}{2 (8)} \cdot 2)$

Cancel the common factor.

$e = \frac{1}{8} - (\frac{9}{2 \cdot 8} \cdot 2)$

Rewrite the expression.

$e = \frac{1}{8} - \frac{9}{8}$

Combine the numerators over the common denominator.

$e = \frac{1 - 9}{8}$

Subtract $9$ from $1$ .

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

Divide $- 8$ by $8$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x + 3)}^{2} - 1$ .

$\frac{1}{8} {(x + 3)}^{2} - 1$

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = \frac{1}{8}$

$h = - 3$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(- 3, - 1)$

Step 4

Find the Vertex x^2+6x-8y+1=0

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