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

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

Step 1

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

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

$a = - 6$

$b = 3$

$c = 2$

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

$d = \frac{3}{2 \cdot - 6}$

Simplify the right side.

Cancel the common factor of $3$ and $- 6$ .

Factor $3$ out of $3$ .

$d = \frac{3 (1)}{2 \cdot - 6}$

Cancel the common factors.

Factor $3$ out of $2 \cdot - 6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 2$ .

$d = \frac{1}{- 4}$

Move the negative in front of the fraction.

$d = - \frac{1}{4}$

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

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

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = 2 - \frac{9}{4 \cdot - 6}$

Multiply $4$ by $- 6$ .

$e = 2 - \frac{9}{- 24}$

Cancel the common factor of $9$ and $- 24$ .

Factor $3$ out of $9$ .

$e = 2 - \frac{3 (3)}{- 24}$

Cancel the common factors.

Factor $3$ out of $- 24$ .

$e = 2 - \frac{3 \cdot 3}{3 \cdot - 8}$

Cancel the common factor.

$e = 2 - \frac{3 \cdot 3}{3 \cdot - 8}$

Rewrite the expression.

$e = 2 - \frac{3}{- 8}$

Move the negative in front of the fraction.

$e = 2 - - \frac{3}{8}$

Multiply $- - \frac{3}{8}$ .

Multiply $- 1$ by $- 1$ .

$e = 2 + 1 (\frac{3}{8})$

Multiply $\frac{3}{8}$ by $1$ .

$e = 2 + \frac{3}{8}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = 2 \cdot \frac{8}{8} + \frac{3}{8}$

Combine $2$ and $\frac{8}{8}$ .

$e = \frac{2 \cdot 8}{8} + \frac{3}{8}$

Combine the numerators over the common denominator.

$e = \frac{2 \cdot 8 + 3}{8}$

Simplify the numerator.

Multiply $2$ by $8$ .

$e = \frac{16 + 3}{8}$

Add $16$ and $3$ .

$e = \frac{19}{8}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 6 {(x - \frac{1}{4})}^{2} + \frac{19}{8}$ .

$- 6 {(x - \frac{1}{4})}^{2} + \frac{19}{8}$

Step 2

Set $y$ equal to the new right side.

$y = - 6 {(x - \frac{1}{4})}^{2} + \frac{19}{8}$

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

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