Find the Vertex Form f(x)=6x^2+5-42x

$f (x) = 6 x^{2} + 5 - 42 x$

Step 1

Write $f (x) = 6 x^{2} + 5 - 42 x$ as an equation.

$y = 6 x^{2} + 5 - 42 x$

Step 2

Move $5$ .

$y = 6 x^{2} - 42 x + 5$

Step 3

Complete the square for $6 x^{2} - 42 x + 5$ .

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

$a = 6$

$b = - 42$

$c = 5$

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

$d = \frac{- 42}{2 \cdot 6}$

Simplify the right side.

Cancel the common factor of $- 42$ and $2$ .

Factor $2$ out of $- 42$ .

$d = \frac{2 \cdot - 21}{2 \cdot 6}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 21}{2 (6)}$

Cancel the common factor.

$d = \frac{2 \cdot - 21}{2 \cdot 6}$

Rewrite the expression.

$d = \frac{- 21}{6}$

Cancel the common factor of $- 21$ and $6$ .

Factor $3$ out of $- 21$ .

$d = \frac{3 (- 7)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$d = \frac{3 \cdot - 7}{3 \cdot 2}$

Cancel the common factor.

$d = \frac{3 \cdot - 7}{3 \cdot 2}$

Rewrite the expression.

$d = \frac{- 7}{2}$

Move the negative in front of the fraction.

$d = - \frac{7}{2}$

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

$e = 5 - \frac{{(- 42)}^{2}}{4 \cdot 6}$

Simplify the right side.

Simplify each term.

Raise $- 42$ to the power of $2$ .

$e = 5 - \frac{1764}{4 \cdot 6}$

Multiply $4$ by $6$ .

$e = 5 - \frac{1764}{24}$

Cancel the common factor of $1764$ and $24$ .

Factor $12$ out of $1764$ .

$e = 5 - \frac{12 (147)}{24}$

Cancel the common factors.

Factor $12$ out of $24$ .

$e = 5 - \frac{12 \cdot 147}{12 \cdot 2}$

Cancel the common factor.

$e = 5 - \frac{12 \cdot 147}{12 \cdot 2}$

Rewrite the expression.

$e = 5 - \frac{147}{2}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 5 \cdot \frac{2}{2} - \frac{147}{2}$

Combine $5$ and $\frac{2}{2}$ .

$e = \frac{5 \cdot 2}{2} - \frac{147}{2}$

Combine the numerators over the common denominator.

$e = \frac{5 \cdot 2 - 147}{2}$

Simplify the numerator.

Multiply $5$ by $2$ .

$e = \frac{10 - 147}{2}$

Subtract $147$ from $10$ .

$e = \frac{- 137}{2}$

Move the negative in front of the fraction.

$e = - \frac{137}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(x - \frac{7}{2})}^{2} - \frac{137}{2}$ .

$6 {(x - \frac{7}{2})}^{2} - \frac{137}{2}$

Step 4

Set $y$ equal to the new right side.

$y = 6 {(x - \frac{7}{2})}^{2} - \frac{137}{2}$

Find the Vertex Form f(x)=6x^2+5-42x

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