Question

write the function in the form f(x)=a(x - h)^2 + k by completing the square. then identify the vertex. express numbers in exact, simplified form.
q(x) = 3x^2 - 18x + 11
part: 0 / 2
part 1 of 2
the function written in the form f(x)=a(x - h)^2 + k is q(x) = .

Explanation:

Step1: Factor out the coefficient of \(x^2\)

First, we factor out the coefficient of \(x^2\) from the first two terms of the quadratic function \(q(x) = 3x^2 - 18x + 11\). The coefficient of \(x^2\) is 3, so we have:
\(q(x)=3(x^2 - 6x)+11\)

Step2: Complete the square inside the parentheses

To complete the square for the expression \(x^2 - 6x\), we take half of the coefficient of \(x\) (which is \(-6\)), square it, and add and subtract that value inside the parentheses.

Half of \(-6\) is \(-3\), and squaring \(-3\) gives \(9\). So we add and subtract 9 inside the parentheses:
\(q(x)=3(x^2 - 6x + 9 - 9)+11\)

Step3: Rewrite the perfect square trinomial

Now, we can rewrite \(x^2 - 6x + 9\) as a perfect square \((x - 3)^2\):
\(q(x)=3((x - 3)^2 - 9)+11\)

Step4: Distribute the 3 and simplify

Next, we distribute the 3 to both terms inside the parentheses:
\(q(x)=3(x - 3)^2 - 27 + 11\)

Then, we simplify the constant terms:
\(q(x)=3(x - 3)^2 - 16\)

Answer:

\(3(x - 3)^2 - 16\)