Find the Vertex g(r)=-(r+11)(r+14)

$g (r) = - (r + 11) (r + 14)$

Step 1

Rewrite the equation in terms of $x$ and $y$ .

$y = - (x + 11) (x + 14)$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- (x + 11) (x + 14)$ .

Simplify the expression.

Apply the distributive property.

$(- x - 1 \cdot 11) (x + 14)$

Multiply $- 1$ by $11$ .

$(- x - 11) (x + 14)$

Expand $(- x - 11) (x + 14)$ using the FOIL Method.

Apply the distributive property.

$- x (x + 14) - 11 (x + 14)$

Apply the distributive property.

$- x \cdot x - x \cdot 14 - 11 (x + 14)$

Apply the distributive property.

$- x \cdot x - x \cdot 14 - 11 x - 11 \cdot 14$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$- (x \cdot x) - x \cdot 14 - 11 x - 11 \cdot 14$

Multiply $x$ by $x$ .

$- x^{2} - x \cdot 14 - 11 x - 11 \cdot 14$

Multiply $14$ by $- 1$ .

$- x^{2} - 14 x - 11 x - 11 \cdot 14$

Multiply $- 11$ by $14$ .

$- x^{2} - 14 x - 11 x - 154$

Subtract $11 x$ from $- 14 x$ .

$- x^{2} - 25 x - 154$

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

$a = - 1$

$b = - 25$

$c = - 154$

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

$d = \frac{- 25}{2 \cdot - 1}$

Simplify the right side.

Multiply $2$ by $- 1$ .

$d = \frac{- 25}{- 2}$

Dividing two negative values results in a positive value.

$d = \frac{25}{2}$

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

$e = - 154 - \frac{{(- 25)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $- 25$ to the power of $2$ .

$e = - 154 - \frac{625}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 154 - \frac{625}{- 4}$

Move the negative in front of the fraction.

$e = - 154 - - \frac{625}{4}$

Multiply $- - \frac{625}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = - 154 + 1 (\frac{625}{4})$

Multiply $\frac{625}{4}$ by $1$ .

$e = - 154 + \frac{625}{4}$

To write $- 154$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 154 \cdot \frac{4}{4} + \frac{625}{4}$

Combine $- 154$ and $\frac{4}{4}$ .

$e = \frac{- 154 \cdot 4}{4} + \frac{625}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 154 \cdot 4 + 625}{4}$

Simplify the numerator.

Multiply $- 154$ by $4$ .

$e = \frac{- 616 + 625}{4}$

Add $- 616$ and $625$ .

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

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + \frac{25}{2})}^{2} + \frac{9}{4}$ .

$- {(x + \frac{25}{2})}^{2} + \frac{9}{4}$

Set $y$ equal to the new right side.

$y = - {(x + \frac{25}{2})}^{2} + \frac{9}{4}$

Step 3

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

$a = - 1$

$h = - \frac{25}{2}$

$k = \frac{9}{4}$

Step 4

Find the vertex $(h, k)$ .

$(- \frac{25}{2}, \frac{9}{4})$

Step 5

Find the Vertex g(r)=-(r+11)(r+14)

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