Find the Vertex 4x^2+24x+41

$4 x^{2} + 24 x + 41$

Step 1

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} + 24 x + 41$ .

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

$a = 4$

$b = 24$

$c = 41$

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

$d = \frac{24}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot 12}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{12}{4}$

Cancel the common factor of $12$ and $4$ .

Factor $4$ out of $12$ .

$d = \frac{4 \cdot 3}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$d = \frac{4 \cdot 3}{4 (1)}$

Cancel the common factor.

$d = \frac{4 \cdot 3}{4 \cdot 1}$

Rewrite the expression.

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

Divide $3$ by $1$ .

$d = 3$

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

$e = 41 - \frac{24^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $24$ to the power of $2$ .

$e = 41 - \frac{576}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 41 - \frac{576}{16}$

Divide $576$ by $16$ .

$e = 41 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 41 - 36$

Subtract $36$ from $41$ .

$e = 5$

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

$4 {(x + 3)}^{2} + 5$

Set $y$ equal to the new right side.

$y = 4 {(x + 3)}^{2} + 5$

Step 2

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

$a = 4$

$h = - 3$

$k = 5$

Step 3

Find the vertex $(h, k)$ .

$(- 3, 5)$

Step 4

Find the Vertex 4x^2+24x+41

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