Find the Vertex f(x)=-4x^2-10x-2

$f (x) = - 4 x^{2} - 10 x - 2$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 4 x^{2} - 10 x - 2$ .

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

$a = - 4$

$b = - 10$

$c = - 2$

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

$d = \frac{- 10}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot - 5}{2 (- 4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{- 5}{- 4}$

Dividing two negative values results in a positive value.

$d = \frac{5}{4}$

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

$e = - 2 - \frac{{(- 10)}^{2}}{4 \cdot - 4}$

Simplify the right side.

Simplify each term.

Raise $- 10$ to the power of $2$ .

$e = - 2 - \frac{100}{4 \cdot - 4}$

Multiply $4$ by $- 4$ .

$e = - 2 - \frac{100}{- 16}$

Cancel the common factor of $100$ and $- 16$ .

Factor $4$ out of $100$ .

$e = - 2 - \frac{4 (25)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$e = - 2 - \frac{4 \cdot 25}{4 \cdot - 4}$

Cancel the common factor.

$e = - 2 - \frac{4 \cdot 25}{4 \cdot - 4}$

Rewrite the expression.

$e = - 2 - \frac{25}{- 4}$

Move the negative in front of the fraction.

$e = - 2 - - \frac{25}{4}$

Multiply $- - \frac{25}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = - 2 + 1 (\frac{25}{4})$

Multiply $\frac{25}{4}$ by $1$ .

$e = - 2 + \frac{25}{4}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 2 \cdot \frac{4}{4} + \frac{25}{4}$

Combine $- 2$ and $\frac{4}{4}$ .

$e = \frac{- 2 \cdot 4}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 4 + 25}{4}$

Simplify the numerator.

Multiply $- 2$ by $4$ .

$e = \frac{- 8 + 25}{4}$

Add $- 8$ and $25$ .

$e = \frac{17}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + \frac{5}{4})}^{2} + \frac{17}{4}$ .

$- 4 {(x + \frac{5}{4})}^{2} + \frac{17}{4}$

Set $y$ equal to the new right side.

$y = - 4 {(x + \frac{5}{4})}^{2} + \frac{17}{4}$

Step 2

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

$a = - 4$

$h = - \frac{5}{4}$

$k = \frac{17}{4}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{5}{4}, \frac{17}{4})$

Step 4

Find the Vertex f(x)=-4x^2-10x-2

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