Find the Vertex 3x^2-30x+78

$3 x^{2} - 30 x + 78$

Step 1

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} - 30 x + 78$ .

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

$a = 3$

$b = - 30$

$c = 78$

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

$e = 78 - \frac{{(- 30)}^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

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

$e = 78 - \frac{900}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 78 - \frac{900}{12}$

Divide $900$ by $12$ .

$e = 78 - 1 \cdot 75$

Multiply $- 1$ by $75$ .

$e = 78 - 75$

Subtract $75$ from $78$ .

$e = 3$

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

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

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 3$

$h = 5$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(5, 3)$

Step 4

Find the Vertex 3x^2-30x+78

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