Find the Vertex 4x^2-32x+65

$4 x^{2} - 32 x + 65$

Step 1

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} - 32 x + 65$ .

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

$a = 4$

$b = - 32$

$c = 65$

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

$d = \frac{- 32}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $- 32$ and $2$ .

Factor $2$ out of $- 32$ .

$d = \frac{2 \cdot - 16}{2 \cdot 4}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 16}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 16}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{- 16}{4}$

Cancel the common factor of $- 16$ and $4$ .

Factor $4$ out of $- 16$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

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

$e = 65 - \frac{{(- 32)}^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $- 32$ to the power of $2$ .

$e = 65 - \frac{1024}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 65 - \frac{1024}{16}$

Divide $1024$ by $16$ .

$e = 65 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 65 - 64$

Subtract $64$ from $65$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 4)}^{2} + 1$ .

$4 {(x - 4)}^{2} + 1$

Set $y$ equal to the new right side.

$y = 4 {(x - 4)}^{2} + 1$

Step 2

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

$a = 4$

$h = 4$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(4, 1)$

Step 4

Find the Vertex 4x^2-32x+65

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