Find the Vertex Form f(x)=-3x^2+15x+71

$f (x) = - 3 x^{2} + 15 x + 71$

Step 1

Write $f (x) = - 3 x^{2} + 15 x + 71$ as an equation.

$y = - 3 x^{2} + 15 x + 71$

Step 2

Complete the square for $- 3 x^{2} + 15 x + 71$ .

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

$a = - 3$

$b = 15$

$c = 71$

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

$d = \frac{15}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $15$ and $- 3$ .

Factor $3$ out of $15$ .

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

Cancel the common factors.

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

$d = \frac{3 (5)}{3 (2 \cdot - 1)}$

Cancel the common factor.

$d = \frac{3 \cdot 5}{3 (2 \cdot - 1)}$

Rewrite the expression.

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

Multiply $2$ by $- 1$ .

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

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

$e = 71 - \frac{15^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $15$ to the power of $2$ .

$e = 71 - \frac{225}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = 71 - \frac{225}{- 12}$

Cancel the common factor of $225$ and $- 12$ .

Factor $3$ out of $225$ .

$e = 71 - \frac{3 (75)}{- 12}$

Cancel the common factors.

Factor $3$ out of $- 12$ .

$e = 71 - \frac{3 \cdot 75}{3 \cdot - 4}$

Cancel the common factor.

$e = 71 - \frac{3 \cdot 75}{3 \cdot - 4}$

Rewrite the expression.

$e = 71 - \frac{75}{- 4}$

Move the negative in front of the fraction.

$e = 71 - - \frac{75}{4}$

Multiply $- - \frac{75}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = 71 + 1 (\frac{75}{4})$

Multiply $\frac{75}{4}$ by $1$ .

$e = 71 + \frac{75}{4}$

To write $71$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 71 \cdot \frac{4}{4} + \frac{75}{4}$

Combine $71$ and $\frac{4}{4}$ .

$e = \frac{71 \cdot 4}{4} + \frac{75}{4}$

Combine the numerators over the common denominator.

$e = \frac{71 \cdot 4 + 75}{4}$

Simplify the numerator.

Multiply $71$ by $4$ .

$e = \frac{284 + 75}{4}$

Add $284$ and $75$ .

$e = \frac{359}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{5}{2})}^{2} + \frac{359}{4}$ .

$- 3 {(x - \frac{5}{2})}^{2} + \frac{359}{4}$

Step 3

Set $y$ equal to the new right side.

$y = - 3 {(x - \frac{5}{2})}^{2} + \frac{359}{4}$

Find the Vertex Form f(x)=-3x^2+15x+71

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