Find the Vertex Form f(x)=2(x^2-22x+121)+185-242

$f (x) = 2 (x^{2} - 22 x + 121) + 185 - 242$

Step 1

Write $f (x) = 2 (x^{2} - 22 x + 121) + 185 - 242$ as an equation.

$y = 2 (x^{2} - 22 x + 121) + 185 - 242$

Step 2

Simplify $2 (x^{2} - 22 x + 121) + 185 - 242$ .

Simplify each term.

Apply the distributive property.

$y = 2 x^{2} + 2 (- 22 x) + 2 \cdot 121 + 185 - 242$

Simplify.

Multiply $- 22$ by $2$ .

$y = 2 x^{2} - 44 x + 2 \cdot 121 + 185 - 242$

Multiply $2$ by $121$ .

$y = 2 x^{2} - 44 x + 242 + 185 - 242$

Combine the opposite terms in $2 x^{2} - 44 x + 242 + 185 - 242$ .

Subtract $242$ from $242$ .

$y = 2 x^{2} - 44 x + 0 + 185$

Add $2 x^{2} - 44 x$ and $0$ .

$y = 2 x^{2} - 44 x + 185$

Step 3

Complete the square for $2 x^{2} - 44 x + 185$ .

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

$a = 2$

$b = - 44$

$c = 185$

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

$d = \frac{- 44}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $- 44$ and $2$ .

Factor $2$ out of $- 44$ .

$d = \frac{2 \cdot - 22}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 22}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 22}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{- 22}{2}$

Cancel the common factor of $- 22$ and $2$ .

Factor $2$ out of $- 22$ .

$d = \frac{2 \cdot - 11}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot - 11}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 11}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 11}{1}$

Divide $- 11$ by $1$ .

$d = - 11$

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

$e = 185 - \frac{{(- 44)}^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $- 44$ to the power of $2$ .

$e = 185 - \frac{1936}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 185 - \frac{1936}{8}$

Divide $1936$ by $8$ .

$e = 185 - 1 \cdot 242$

Multiply $- 1$ by $242$ .

$e = 185 - 242$

Subtract $242$ from $185$ .

$e = - 57$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - 11)}^{2} - 57$ .

$2 {(x - 11)}^{2} - 57$

Step 4

Set $y$ equal to the new right side.

$y = 2 {(x - 11)}^{2} - 57$

Find the Vertex Form f(x)=2(x^2-22x+121)+185-242

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