Find the Vertex -1/5x^2-2/11x+1/121

$- \frac{1}{5} x^{2} - \frac{2}{11} x + \frac{1}{121}$

Step 1

Rewrite the equation in vertex form.

Simplify each term.

Combine $x^{2}$ and $\frac{1}{5}$ .

$y = - \frac{x^{2}}{5} - \frac{2}{11} x + \frac{1}{121}$

Combine $x$ and $\frac{2}{11}$ .

$y = - \frac{x^{2}}{5} - \frac{x \cdot 2}{11} + \frac{1}{121}$

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

$y = - \frac{x^{2}}{5} - \frac{2 x}{11} + \frac{1}{121}$

Complete the square for $- \frac{x^{2}}{5} - \frac{2 x}{11} + \frac{1}{121}$ .

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

$a = - \frac{1}{5}$

$b = - \frac{2}{11}$

$c = \frac{1}{121}$

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

$d = \frac{- \frac{2}{11}}{2 (- \frac{1}{5})}$

Simplify the right side.

Dividing two negative values results in a positive value.

$d = \frac{\frac{2}{11}}{2 (\frac{1}{5})}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{2}{11} \cdot \frac{1}{2 (\frac{1}{5})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{11} \cdot \frac{1}{2 (\frac{1}{5})}$

Rewrite the expression.

$d = \frac{1}{11} \cdot \frac{1}{\frac{1}{5}}$

Multiply $\frac{1}{11}$ by $\frac{1}{\frac{1}{5}}$ .

$d = \frac{1}{11 (\frac{1}{5})}$

Combine $11$ and $\frac{1}{5}$ .

$d = \frac{1}{\frac{11}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 1 (\frac{5}{11})$

Multiply $\frac{5}{11}$ by $1$ .

$d = \frac{5}{11}$

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

$e = \frac{1}{121} - \frac{{(- \frac{2}{11})}^{2}}{4 (- \frac{1}{5})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{2}{11}$ .

$e = \frac{1}{121} - \frac{{(- 1)}^{2} {(\frac{2}{11})}^{2}}{4 (- \frac{1}{5})}$

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

$e = \frac{1}{121} - \frac{1 {(\frac{2}{11})}^{2}}{4 (- \frac{1}{5})}$

Apply the product rule to $\frac{2}{11}$ .

$e = \frac{1}{121} - \frac{1 \frac{2^{2}}{11^{2}}}{4 (- \frac{1}{5})}$

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

$e = \frac{1}{121} - \frac{1 \frac{4}{11^{2}}}{4 (- \frac{1}{5})}$

Raise $11$ to the power of $2$ .

$e = \frac{1}{121} - \frac{1 (\frac{4}{121})}{4 (- \frac{1}{5})}$

Multiply $\frac{4}{121}$ by $1$ .

$e = \frac{1}{121} - \frac{\frac{4}{121}}{4 (- \frac{1}{5})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{1}{121} - \frac{\frac{4}{121}}{- 4 (\frac{1}{5})}$

Combine $- 4$ and $\frac{1}{5}$ .

$e = \frac{1}{121} - \frac{\frac{4}{121}}{\frac{- 4}{5}}$

Move the negative in front of the fraction.

$e = \frac{1}{121} - \frac{\frac{4}{121}}{- \frac{4}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{121} - (\frac{4}{121} (- \frac{5}{4}))$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$e = \frac{1}{121} - (\frac{4}{121} \cdot \frac{- 5}{4})$

Cancel the common factor.

$e = \frac{1}{121} - (\frac{4}{121} \cdot \frac{- 5}{4})$

Rewrite the expression.

$e = \frac{1}{121} - (\frac{1}{121} \cdot - 5)$

Combine $\frac{1}{121}$ and $- 5$ .

$e = \frac{1}{121} - \frac{- 5}{121}$

Move the negative in front of the fraction.

$e = \frac{1}{121} - - \frac{5}{121}$

Multiply $- - \frac{5}{121}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{1}{121} + 1 (\frac{5}{121})$

Multiply $\frac{5}{121}$ by $1$ .

$e = \frac{1}{121} + \frac{5}{121}$

Combine the numerators over the common denominator.

$e = \frac{1 + 5}{121}$

Add $1$ and $5$ .

$e = \frac{6}{121}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{5} {(x + \frac{5}{11})}^{2} + \frac{6}{121}$ .

$- \frac{1}{5} {(x + \frac{5}{11})}^{2} + \frac{6}{121}$

Set $y$ equal to the new right side.

$y = - \frac{1}{5} \cdot {(x + \frac{5}{11})}^{2} + \frac{6}{121}$

Step 2

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

$a = - \frac{1}{5}$

$h = - \frac{5}{11}$

$k = \frac{6}{121}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{5}{11}, \frac{6}{121})$

Step 4

Find the Vertex -1/5x^2-2/11x+1/121

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