Find the Vertex 8x^2-64x

$8 x^{2} - 64 x$

Step 1

Rewrite the equation in vertex form.

Complete the square for $8 x^{2} - 64 x$ .

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

$a = 8$

$b = - 64$

$c = 0$

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

$d = \frac{- 64}{2 \cdot 8}$

Simplify the right side.

Cancel the common factor of $- 64$ and $2$ .

Factor $2$ out of $- 64$ .

$d = \frac{2 \cdot - 32}{2 \cdot 8}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 8$ .

$d = \frac{2 \cdot - 32}{2 (8)}$

Cancel the common factor.

$d = \frac{2 \cdot - 32}{2 \cdot 8}$

Rewrite the expression.

$d = \frac{- 32}{8}$

Cancel the common factor of $- 32$ and $8$ .

Factor $8$ out of $- 32$ .

$d = \frac{8 \cdot - 4}{8}$

Cancel the common factors.

Factor $8$ out of $8$ .

$d = \frac{8 \cdot - 4}{8 (1)}$

Cancel the common factor.

$d = \frac{8 \cdot - 4}{8 \cdot 1}$

Rewrite the expression.

$d = \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = 0 - \frac{{(- 64)}^{2}}{4 \cdot 8}$

Simplify the right side.

Simplify each term.

Raise $- 64$ to the power of $2$ .

$e = 0 - \frac{4096}{4 \cdot 8}$

Multiply $4$ by $8$ .

$e = 0 - \frac{4096}{32}$

Divide $4096$ by $32$ .

$e = 0 - 1 \cdot 128$

Multiply $- 1$ by $128$ .

$e = 0 - 128$

Subtract $128$ from $0$ .

$e = - 128$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $8 {(x - 4)}^{2} - 128$ .

$8 {(x - 4)}^{2} - 128$

Set $y$ equal to the new right side.

$y = 8 {(x - 4)}^{2} - 128$

Step 2

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

$a = 8$

$h = 4$

$k = - 128$

Step 3

Find the vertex $(h, k)$ .

$(4, - 128)$

Step 4

Find the Vertex 8x^2-64x

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