Simplify f(x) = square root of 2x^2-36x+162

$f (x) = \sqrt{2 x^{2} - 36 x + 162}$

Step 1

Factor $2$ out of $2 x^{2} - 36 x + 162$ .

Factor $2$ out of $2 x^{2}$ .

$f (x) = \sqrt{2 (x^{2}) - 36 x + 162}$

Factor $2$ out of $- 36 x$ .

$f (x) = \sqrt{2 (x^{2}) + 2 (- 18 x) + 162}$

Factor $2$ out of $162$ .

$f (x) = \sqrt{2 x^{2} + 2 (- 18 x) + 2 \cdot 81}$

Factor $2$ out of $2 x^{2} + 2 (- 18 x)$ .

$f (x) = \sqrt{2 (x^{2} - 18 x) + 2 \cdot 81}$

Factor $2$ out of $2 (x^{2} - 18 x) + 2 \cdot 81$ .

$f (x) = \sqrt{2 (x^{2} - 18 x + 81)}$

Step 2

Factor using the perfect square rule.

Rewrite $81$ as $9^{2}$ .

$f (x) = \sqrt{2 (x^{2} - 18 x + 9^{2})}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 x = 2 \cdot x \cdot 9$

Rewrite the polynomial.

$f (x) = \sqrt{2 (x^{2} - 2 \cdot x \cdot 9 + 9^{2})}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

$f (x) = \sqrt{2 {(x - 9)}^{2}}$

Step 3

Simplify by cancelling exponent with radical.

Reorder $2$ and ${(x - 9)}^{2}$ .

$f (x) = \sqrt{{(x - 9)}^{2} \cdot 2}$

Pull terms out from under the radical.

$f (x) = (x - 9) \sqrt{2}$