Find the Vertex y=-3x^2+5x-6

$y = - 3 x^{2} + 5 x - 6$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 3 x^{2} + 5 x - 6$ .

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

$a = - 3$

$b = 5$

$c = - 6$

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

$d = \frac{5}{2 \cdot - 3}$

Simplify the right side.

Multiply $2$ by $- 3$ .

$d = \frac{5}{- 6}$

Move the negative in front of the fraction.

$d = - \frac{5}{6}$

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

$e = - 6 - \frac{5^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $5$ to the power of $2$ .

$e = - 6 - \frac{25}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 6 - \frac{25}{- 12}$

Move the negative in front of the fraction.

$e = - 6 - - \frac{25}{12}$

Multiply $- - \frac{25}{12}$ .

Multiply $- 1$ by $- 1$ .

$e = - 6 + 1 (\frac{25}{12})$

Multiply $\frac{25}{12}$ by $1$ .

$e = - 6 + \frac{25}{12}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$e = - 6 \cdot \frac{12}{12} + \frac{25}{12}$

Combine $- 6$ and $\frac{12}{12}$ .

$e = \frac{- 6 \cdot 12}{12} + \frac{25}{12}$

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 12 + 25}{12}$

Simplify the numerator.

Multiply $- 6$ by $12$ .

$e = \frac{- 72 + 25}{12}$

Add $- 72$ and $25$ .

$e = \frac{- 47}{12}$

Move the negative in front of the fraction.

$e = - \frac{47}{12}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{5}{6})}^{2} - \frac{47}{12}$ .

$- 3 {(x - \frac{5}{6})}^{2} - \frac{47}{12}$

Set $y$ equal to the new right side.

$y = - 3 {(x - \frac{5}{6})}^{2} - \frac{47}{12}$

Step 2

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

$a = - 3$

$h = \frac{5}{6}$

$k = - \frac{47}{12}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{5}{6}, - \frac{47}{12})$

Step 4

Find the Vertex y=-3x^2+5x-6

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