Find the Vertex (x-2)(x-6)

$(x - 2) (x - 6)$

Step 1

Rewrite the equation in vertex form.

Complete the square for $(x - 2) (x - 6)$ .

Simplify the expression.

Expand $(x - 2) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$x (x - 6) - 2 (x - 6)$

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 2 (x - 6)$

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 2 x - 2 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 6 - 2 x - 2 \cdot - 6$

Move $- 6$ to the left of $x$ .

$x^{2} - 6 \cdot x - 2 x - 2 \cdot - 6$

Multiply $- 2$ by $- 6$ .

$x^{2} - 6 x - 2 x + 12$

Subtract $2 x$ from $- 6 x$ .

$x^{2} - 8 x + 12$

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

$a = 1$

$b = - 8$

$c = 12$

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

$d = \frac{- 8}{2 \cdot 1}$

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 4}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

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

$e = 12 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$e = 12 - \frac{64}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 12 - \frac{64}{4}$

Divide $64$ by $4$ .

$e = 12 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$e = 12 - 16$

Subtract $16$ from $12$ .

$e = - 4$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

$a = 1$

$h = 4$

$k = - 4$

Step 3

Find the vertex $(h, k)$ .

$(4, - 4)$

Step 4

Find the Vertex (x-2)(x-6)

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