Find the Vertex 8x^2-48x+65

$8 x^{2} - 48 x + 65$

Step 1

Rewrite the equation in vertex form.

Complete the square for $8 x^{2} - 48 x + 65$ .

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

$a = 8$

$b = - 48$

$c = 65$

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

$d = \frac{- 48}{2 \cdot 8}$

Simplify the right side.

Cancel the common factor of $- 48$ and $2$ .

Factor $2$ out of $- 48$ .

$d = \frac{2 \cdot - 24}{2 \cdot 8}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 24}{2 (8)}$

Cancel the common factor.

$d = \frac{2 \cdot - 24}{2 \cdot 8}$

Rewrite the expression.

$d = \frac{- 24}{8}$

Cancel the common factor of $- 24$ and $8$ .

Factor $8$ out of $- 24$ .

$d = \frac{8 \cdot - 3}{8}$

Cancel the common factors.

Factor $8$ out of $8$ .

$d = \frac{8 \cdot - 3}{8 (1)}$

Cancel the common factor.

$d = \frac{8 \cdot - 3}{8 \cdot 1}$

Rewrite the expression.

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

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 65 - \frac{{(- 48)}^{2}}{4 \cdot 8}$

Simplify the right side.

Simplify each term.

Raise $- 48$ to the power of $2$ .

$e = 65 - \frac{2304}{4 \cdot 8}$

Multiply $4$ by $8$ .

$e = 65 - \frac{2304}{32}$

Divide $2304$ by $32$ .

$e = 65 - 1 \cdot 72$

Multiply $- 1$ by $72$ .

$e = 65 - 72$

Subtract $72$ from $65$ .

$e = - 7$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $8 {(x - 3)}^{2} - 7$ .

$8 {(x - 3)}^{2} - 7$

Set $y$ equal to the new right side.

$y = 8 {(x - 3)}^{2} - 7$

Step 2

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

$a = 8$

$h = 3$

$k = - 7$

Step 3

Find the vertex $(h, k)$ .

$(3, - 7)$

Step 4

Find the Vertex 8x^2-48x+65

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