Find the Focus y=2x^2+11x+5

$y = 2 x^{2} + 11 x + 5$

Step 1

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} + 11 x + 5$ .

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

$a = 2$

$b = 11$

$c = 5$

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

$d = \frac{11}{2 \cdot 2}$

Multiply $2$ by $2$ .

$d = \frac{11}{4}$

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

$e = 5 - \frac{11^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

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

$e = 5 - \frac{121}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 5 - \frac{121}{8}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = 5 \cdot \frac{8}{8} - \frac{121}{8}$

Combine $5$ and $\frac{8}{8}$ .

$e = \frac{5 \cdot 8}{8} - \frac{121}{8}$

Combine the numerators over the common denominator.

$e = \frac{5 \cdot 8 - 121}{8}$

Simplify the numerator.

Multiply $5$ by $8$ .

$e = \frac{40 - 121}{8}$

Subtract $121$ from $40$ .

$e = \frac{- 81}{8}$

Move the negative in front of the fraction.

$e = - \frac{81}{8}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{11}{4})}^{2} - \frac{81}{8}$ .

$2 {(x + \frac{11}{4})}^{2} - \frac{81}{8}$

Set $y$ equal to the new right side.

$y = 2 {(x + \frac{11}{4})}^{2} - \frac{81}{8}$

Step 2

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

$a = 2$

$h = - \frac{11}{4}$

$k = - \frac{81}{8}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{11}{4}, - \frac{81}{8})$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 2}$

Multiply $4$ by $2$ .

$\frac{1}{8}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- \frac{11}{4}, - 10)$

Step 6

Find the Focus y=2x^2+11x+5

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