Find the Vertex Form f(x)=3x^2+30x+74

$f (x) = 3 x^{2} + 30 x + 74$

Step 1

Write $f (x) = 3 x^{2} + 30 x + 74$ as an equation.

$y = 3 x^{2} + 30 x + 74$

Step 2

Complete the square for $3 x^{2} + 30 x + 74$ .

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

$a = 3$

$b = 30$

$c = 74$

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

$d = \frac{30}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $30$ and $2$ .

Factor $2$ out of $30$ .

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

Cancel the common factors.

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

$d = \frac{2 \cdot 15}{2 (3)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{15}{3}$

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

Factor $3$ out of $15$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

$d = \frac{3 \cdot 5}{3 \cdot 1}$

Rewrite the expression.

$d = \frac{5}{1}$

Divide $5$ by $1$ .

$d = 5$

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

$e = 74 - \frac{30^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $30$ to the power of $2$ .

$e = 74 - \frac{900}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 74 - \frac{900}{12}$

Divide $900$ by $12$ .

$e = 74 - 1 \cdot 75$

Multiply $- 1$ by $75$ .

$e = 74 - 75$

Subtract $75$ from $74$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 5)}^{2} - 1$ .

$3 {(x + 5)}^{2} - 1$

Step 3

Set $y$ equal to the new right side.

$y = 3 {(x + 5)}^{2} - 1$

Find the Vertex Form f(x)=3x^2+30x+74

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