Find the Vertex -2x^2+10x-7

$- 2 x^{2} + 10 x - 7$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} + 10 x - 7$ .

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

$a = - 2$

$b = 10$

$c = - 7$

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

$d = \frac{10}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

$e = - 7 - \frac{10^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $10$ to the power of $2$ .

$e = - 7 - \frac{100}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 7 - \frac{100}{- 8}$

Cancel the common factor of $100$ and $- 8$ .

Factor $4$ out of $100$ .

$e = - 7 - \frac{4 (25)}{- 8}$

Cancel the common factors.

Factor $4$ out of $- 8$ .

$e = - 7 - \frac{4 \cdot 25}{4 \cdot - 2}$

Cancel the common factor.

$e = - 7 - \frac{4 \cdot 25}{4 \cdot - 2}$

Rewrite the expression.

$e = - 7 - \frac{25}{- 2}$

Move the negative in front of the fraction.

$e = - 7 - - \frac{25}{2}$

Multiply $- - \frac{25}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = - 7 + 1 (\frac{25}{2})$

Multiply $\frac{25}{2}$ by $1$ .

$e = - 7 + \frac{25}{2}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 7 \cdot \frac{2}{2} + \frac{25}{2}$

Combine $- 7$ and $\frac{2}{2}$ .

$e = \frac{- 7 \cdot 2}{2} + \frac{25}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 7 \cdot 2 + 25}{2}$

Simplify the numerator.

Multiply $- 7$ by $2$ .

$e = \frac{- 14 + 25}{2}$

Add $- 14$ and $25$ .

$e = \frac{11}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - \frac{5}{2})}^{2} + \frac{11}{2}$ .

$- 2 {(x - \frac{5}{2})}^{2} + \frac{11}{2}$

Set $y$ equal to the new right side.

$y = - 2 {(x - \frac{5}{2})}^{2} + \frac{11}{2}$

Step 2

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

$a = - 2$

$h = \frac{5}{2}$

$k = \frac{11}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{5}{2}, \frac{11}{2})$

Step 4

Find the Vertex -2x^2+10x-7

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