Find the Vertex Form p(x)=-7x^2+21x

$p (x) = - 7 x^{2} + 21 x$

Step 1

Write $p (x) = - 7 x^{2} + 21 x$ as an equation.

$y = - 7 x^{2} + 21 x$

Step 2

Complete the square for $- 7 x^{2} + 21 x$ .

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

$a = - 7$

$b = 21$

$c = 0$

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

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

Simplify the right side.

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

Factor $7$ out of $21$ .

$d = \frac{7 (3)}{2 \cdot - 7}$

Cancel the common factors.

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

$d = \frac{7 (3)}{7 (2 \cdot - 1)}$

Cancel the common factor.

$d = \frac{7 \cdot 3}{7 (2 \cdot - 1)}$

Rewrite the expression.

$d = \frac{3}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$d = \frac{3}{- 2}$

Move the negative in front of the fraction.

$d = - \frac{3}{2}$

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

$e = 0 - \frac{21^{2}}{4 \cdot - 7}$

Simplify the right side.

Simplify each term.

Raise $21$ to the power of $2$ .

$e = 0 - \frac{441}{4 \cdot - 7}$

Multiply $4$ by $- 7$ .

$e = 0 - \frac{441}{- 28}$

Cancel the common factor of $441$ and $- 28$ .

Factor $7$ out of $441$ .

$e = 0 - \frac{7 (63)}{- 28}$

Cancel the common factors.

Factor $7$ out of $- 28$ .

$e = 0 - \frac{7 \cdot 63}{7 \cdot - 4}$

Cancel the common factor.

$e = 0 - \frac{7 \cdot 63}{7 \cdot - 4}$

Rewrite the expression.

$e = 0 - \frac{63}{- 4}$

Move the negative in front of the fraction.

$e = 0 - - \frac{63}{4}$

Multiply $- - \frac{63}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{63}{4})$

Multiply $\frac{63}{4}$ by $1$ .

$e = 0 + \frac{63}{4}$

Add $0$ and $\frac{63}{4}$ .

$e = \frac{63}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 7 {(x - \frac{3}{2})}^{2} + \frac{63}{4}$ .

$- 7 {(x - \frac{3}{2})}^{2} + \frac{63}{4}$

Step 3

Set $y$ equal to the new right side.

$y = - 7 {(x - \frac{3}{2})}^{2} + \frac{63}{4}$

Find the Vertex Form p(x)=-7x^2+21x

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