Find the Vertex y=-x^2+10x-21

$y = - x^{2} + 10 x - 21$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- x^{2} + 10 x - 21$ .

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

$a = - 1$

$b = 10$

$c = - 21$

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

$d = \frac{10}{2 \cdot - 1}$

Simplify the right side.

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

Factor $2$ out of $10$ .

$d = \frac{2 \cdot 5}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$d = - 5$

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

$e = - 21 - \frac{10^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

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

$e = - 21 - \frac{100}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 21 - \frac{100}{- 4}$

Divide $100$ by $- 4$ .

$e = - 21 - - 25$

Multiply $- 1$ by $- 25$ .

$e = - 21 + 25$

Add $- 21$ and $25$ .

$e = 4$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

$a = - 1$

$h = 5$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(5, 4)$

Step 4

Find the Vertex y=-x^2+10x-21

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