Find the Vertex -3x^2+9x+23

$- 3 x^{2} + 9 x + 23$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 3 x^{2} + 9 x + 23$ .

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

$a = - 3$

$b = 9$

$c = 23$

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

$d = \frac{9}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $9$ and $- 3$ .

Factor $3$ out of $9$ .

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

Cancel the common factors.

Factor $3$ out of $2 \cdot - 3$ .

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

Cancel the common factor.

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

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

$e = 23 - \frac{9^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $9$ to the power of $2$ .

$e = 23 - \frac{81}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = 23 - \frac{81}{- 12}$

Cancel the common factor of $81$ and $- 12$ .

Factor $3$ out of $81$ .

$e = 23 - \frac{3 (27)}{- 12}$

Cancel the common factors.

Factor $3$ out of $- 12$ .

$e = 23 - \frac{3 \cdot 27}{3 \cdot - 4}$

Cancel the common factor.

$e = 23 - \frac{3 \cdot 27}{3 \cdot - 4}$

Rewrite the expression.

$e = 23 - \frac{27}{- 4}$

Move the negative in front of the fraction.

$e = 23 - - \frac{27}{4}$

Multiply $- - \frac{27}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = 23 + 1 (\frac{27}{4})$

Multiply $\frac{27}{4}$ by $1$ .

$e = 23 + \frac{27}{4}$

To write $23$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 23 \cdot \frac{4}{4} + \frac{27}{4}$

Combine $23$ and $\frac{4}{4}$ .

$e = \frac{23 \cdot 4}{4} + \frac{27}{4}$

Combine the numerators over the common denominator.

$e = \frac{23 \cdot 4 + 27}{4}$

Simplify the numerator.

Multiply $23$ by $4$ .

$e = \frac{92 + 27}{4}$

Add $92$ and $27$ .

$e = \frac{119}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{3}{2})}^{2} + \frac{119}{4}$ .

$- 3 {(x - \frac{3}{2})}^{2} + \frac{119}{4}$

Set $y$ equal to the new right side.

$y = - 3 {(x - \frac{3}{2})}^{2} + \frac{119}{4}$

Step 2

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

$a = - 3$

$h = \frac{3}{2}$

$k = \frac{119}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{119}{4})$

Step 4

Find the Vertex -3x^2+9x+23

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