Find the Vertex f(x)=-(x+8)(x-14)

$f (x) = - (x + 8) (x - 14)$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- (x + 8) (x - 14)$ .

Simplify the expression.

Apply the distributive property.

$(- x - 1 \cdot 8) (x - 14)$

Multiply $- 1$ by $8$ .

$(- x - 8) (x - 14)$

Expand $(- x - 8) (x - 14)$ using the FOIL Method.

Apply the distributive property.

$- x (x - 14) - 8 (x - 14)$

Apply the distributive property.

$- x \cdot x - x \cdot - 14 - 8 (x - 14)$

Apply the distributive property.

$- x \cdot x - x \cdot - 14 - 8 x - 8 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$- (x \cdot x) - x \cdot - 14 - 8 x - 8 \cdot - 14$

Multiply $x$ by $x$ .

$- x^{2} - x \cdot - 14 - 8 x - 8 \cdot - 14$

Multiply $- 14$ by $- 1$ .

$- x^{2} + 14 x - 8 x - 8 \cdot - 14$

Multiply $- 8$ by $- 14$ .

$- x^{2} + 14 x - 8 x + 112$

Subtract $8 x$ from $14 x$ .

$- x^{2} + 6 x + 112$

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

$a = - 1$

$b = 6$

$c = 112$

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

$d = \frac{6}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

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

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$d = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$d = - 3$

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

$e = 112 - \frac{6^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = 112 - \frac{36}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 112 - \frac{36}{- 4}$

Divide $36$ by $- 4$ .

$e = 112 - - 9$

Multiply $- 1$ by $- 9$ .

$e = 112 + 9$

Add $112$ and $9$ .

$e = 121$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 3)}^{2} + 121$ .

$- {(x - 3)}^{2} + 121$

Set $y$ equal to the new right side.

$y = - {(x - 3)}^{2} + 121$

Step 2

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

$a = - 1$

$h = 3$

$k = 121$

Step 3

Find the vertex $(h, k)$ .

$(3, 121)$

Step 4

Find the Vertex f(x)=-(x+8)(x-14)

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