Find the Vertex f(x)=2x^2+14x+22

$f (x) = 2 x^{2} + 14 x + 22$

Step 1

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} + 14 x + 22$ .

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

$a = 2$

$b = 14$

$c = 22$

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

$d = \frac{14}{2 \cdot 2}$

Cancel the common factor of $14$ and $2$ .

Factor $2$ out of $14$ .

$d = \frac{2 \cdot 7}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot 7}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot 7}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{7}{2}$

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

$e = 22 - \frac{14^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $14$ to the power of $2$ .

$e = 22 - \frac{196}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 22 - \frac{196}{8}$

Cancel the common factor of $196$ and $8$ .

Factor $4$ out of $196$ .

$e = 22 - \frac{4 (49)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = 22 - \frac{4 \cdot 49}{4 \cdot 2}$

Cancel the common factor.

$e = 22 - \frac{4 \cdot 49}{4 \cdot 2}$

Rewrite the expression.

$e = 22 - \frac{49}{2}$

To write $22$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 22 \cdot \frac{2}{2} - \frac{49}{2}$

Combine $22$ and $\frac{2}{2}$ .

$e = \frac{22 \cdot 2}{2} - \frac{49}{2}$

Combine the numerators over the common denominator.

$e = \frac{22 \cdot 2 - 49}{2}$

Simplify the numerator.

Multiply $22$ by $2$ .

$e = \frac{44 - 49}{2}$

Subtract $49$ from $44$ .

$e = \frac{- 5}{2}$

Move the negative in front of the fraction.

$e = - \frac{5}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{7}{2})}^{2} - \frac{5}{2}$ .

$2 {(x + \frac{7}{2})}^{2} - \frac{5}{2}$

Set $y$ equal to the new right side.

$y = 2 {(x + \frac{7}{2})}^{2} - \frac{5}{2}$

Step 2

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

$a = 2$

$h = - \frac{7}{2}$

$k = - \frac{5}{2}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{7}{2}, - \frac{5}{2})$

Step 4

Find the Vertex f(x)=2x^2+14x+22

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