Find the Vertex Form f(x)=3x^2-14x+10

$f (x) = 3 x^{2} - 14 x + 10$

Step 1

Write $f (x) = 3 x^{2} - 14 x + 10$ as an equation.

$y = 3 x^{2} - 14 x + 10$

Step 2

Complete the square for $3 x^{2} - 14 x + 10$ .

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

$a = 3$

$b = - 14$

$c = 10$

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

$d = \frac{- 14}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

$e = 10 - \frac{{(- 14)}^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $- 14$ to the power of $2$ .

$e = 10 - \frac{196}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 10 - \frac{196}{12}$

Cancel the common factor of $196$ and $12$ .

Factor $4$ out of $196$ .

$e = 10 - \frac{4 (49)}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

$e = 10 - \frac{4 \cdot 49}{4 \cdot 3}$

Cancel the common factor.

$e = 10 - \frac{4 \cdot 49}{4 \cdot 3}$

Rewrite the expression.

$e = 10 - \frac{49}{3}$

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e = 10 \cdot \frac{3}{3} - \frac{49}{3}$

Combine $10$ and $\frac{3}{3}$ .

$e = \frac{10 \cdot 3}{3} - \frac{49}{3}$

Combine the numerators over the common denominator.

$e = \frac{10 \cdot 3 - 49}{3}$

Simplify the numerator.

Multiply $10$ by $3$ .

$e = \frac{30 - 49}{3}$

Subtract $49$ from $30$ .

$e = \frac{- 19}{3}$

Move the negative in front of the fraction.

$e = - \frac{19}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{7}{3})}^{2} - \frac{19}{3}$ .

$3 {(x - \frac{7}{3})}^{2} - \frac{19}{3}$

Step 3

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{7}{3})}^{2} - \frac{19}{3}$

Find the Vertex Form f(x)=3x^2-14x+10

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