Find the Vertex Form f(x)=7x^2+42x

$f (x) = 7 x^{2} + 42 x$

Step 1

Write $f (x) = 7 x^{2} + 42 x$ as an equation.

$y = 7 x^{2} + 42 x$

Step 2

Complete the square for $7 x^{2} + 42 x$ .

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

$a = 7$

$b = 42$

$c = 0$

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

$d = \frac{42}{2 \cdot 7}$

Simplify the right side.

Cancel the common factor of $42$ and $2$ .

Factor $2$ out of $42$ .

$d = \frac{2 \cdot 21}{2 \cdot 7}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 7$ .

$d = \frac{2 \cdot 21}{2 (7)}$

Cancel the common factor.

$d = \frac{2 \cdot 21}{2 \cdot 7}$

Rewrite the expression.

$d = \frac{21}{7}$

Cancel the common factor of $21$ and $7$ .

Factor $7$ out of $21$ .

$d = \frac{7 \cdot 3}{7}$

Cancel the common factors.

Factor $7$ out of $7$ .

$d = \frac{7 \cdot 3}{7 (1)}$

Cancel the common factor.

$d = \frac{7 \cdot 3}{7 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{42^{2}}{4 \cdot 7}$

Simplify the right side.

Simplify each term.

Raise $42$ to the power of $2$ .

$e = 0 - \frac{1764}{4 \cdot 7}$

Multiply $4$ by $7$ .

$e = 0 - \frac{1764}{28}$

Divide $1764$ by $28$ .

$e = 0 - 1 \cdot 63$

Multiply $- 1$ by $63$ .

$e = 0 - 63$

Subtract $63$ from $0$ .

$e = - 63$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $7 {(x + 3)}^{2} - 63$ .

$7 {(x + 3)}^{2} - 63$

Step 3

Set $y$ equal to the new right side.

$y = 7 {(x + 3)}^{2} - 63$

Find the Vertex Form f(x)=7x^2+42x

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