Find the Vertex Form 40x+5x^2

$40 x + 5 x^{2}$

Step 1

Write $40 x + 5 x^{2}$ as an equation.

$y = 40 x + 5 x^{2}$

Step 2

Reorder $40 x$ and $5 x^{2}$ .

$y = 5 x^{2} + 40 x$

Step 3

Complete the square for $5 x^{2} + 40 x$ .

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

$a = 5$

$b = 40$

$c = 0$

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

$d = \frac{40}{2 \cdot 5}$

Simplify the right side.

Cancel the common factor of $40$ and $2$ .

Factor $2$ out of $40$ .

$d = \frac{2 \cdot 20}{2 \cdot 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 5$ .

$d = \frac{2 \cdot 20}{2 (5)}$

Cancel the common factor.

$d = \frac{2 \cdot 20}{2 \cdot 5}$

Rewrite the expression.

$d = \frac{20}{5}$

Cancel the common factor of $20$ and $5$ .

Factor $5$ out of $20$ .

$d = \frac{5 \cdot 4}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$d = \frac{5 \cdot 4}{5 (1)}$

Cancel the common factor.

$d = \frac{5 \cdot 4}{5 \cdot 1}$

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

$e = 0 - \frac{40^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

Raise $40$ to the power of $2$ .

$e = 0 - \frac{1600}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = 0 - \frac{1600}{20}$

Divide $1600$ by $20$ .

$e = 0 - 1 \cdot 80$

Multiply $- 1$ by $80$ .

$e = 0 - 80$

Subtract $80$ from $0$ .

$e = - 80$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + 4)}^{2} - 80$ .

$5 {(x + 4)}^{2} - 80$

Step 4

Set $y$ equal to the new right side.

$y = 5 {(x + 4)}^{2} - 80$

Find the Vertex Form 40x+5x^2

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