Find the Vertex 7x^2-7x

$7 x^{2} - 7 x$

Step 1

Rewrite the equation in vertex form.

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

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

$a = 7$

$b = - 7$

$c = 0$

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

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

Simplify the right side.

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

Factor $7$ out of $- 7$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Simplify the right side.

Cancel the common factor of ${(- 7)}^{2}$ and $7$ .

Rewrite $- 7$ as $- 1 (7)$ .

$e = 0 - \frac{{(- 1 (7))}^{2}}{4 \cdot 7}$

Apply the product rule to $- 1 (7)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 7^{2}}{4 \cdot 7}$

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 \cdot 7^{2}}{4 \cdot 7}$

Multiply $7^{2}$ by $1$ .

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

Factor $7$ out of $7^{2}$ .

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

Cancel the common factors.

Factor $7$ out of $4 \cdot 7$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $\frac{7}{4}$ from $0$ .

$e = - \frac{7}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $7 {(x - \frac{1}{2})}^{2} - \frac{7}{4}$ .

$7 {(x - \frac{1}{2})}^{2} - \frac{7}{4}$

Set $y$ equal to the new right side.

$y = 7 {(x - \frac{1}{2})}^{2} - \frac{7}{4}$

Step 2

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

$a = 7$

$h = \frac{1}{2}$

$k = - \frac{7}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{7}{4})$

Step 4

Find the Vertex 7x^2-7x

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