Questions: Find the vertex form of the quadratic function by completing the square. f(x)=x^2+12x f(x)= (Simplify your answer.)

Transcript text: Find the vertex form of the quadratic function by completing the square. \[ f(x)=x^{2}+12 x \] $f(x)=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Step 1: Identify the Coefficients

The given quadratic function is $ f(x) = x^2 + 12x $. Here, the coefficients are:

$ a = 1 $
$ b = 12 $
$ c = 0 $

Step 2: Calculate the Vertex

To convert the quadratic function into vertex form, we use the formula for the vertex $(h, k)$:

$ h = -\frac{b}{2a} = -\frac{12}{2 \times 1} = -6 $
$ k = c - \frac{b^2}{4a} = 0 - \frac{12^2}{4 \times 1} = -36 $

Step 3: Write the Vertex Form

The vertex form of a quadratic function is given by: \[ f(x) = a(x - h)^2 + k \] Substituting the values of $ a $, $ h $, and $ k $, we get: \[ f(x) = 1(x - (-6))^2 + (-36) \] \[ f(x) = (x + 6)^2 - 36 \]