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To solve the problem, we need to rewrite the quadratic expression \(3x^2 - 12x - 11\) in the form \(3(x + a)^2 + b\).

First, we rewrite the quadratic by completing the square:

1. Factor out the 3 from the first two terms: \(3(x^2 - 4x)\).
2. To complete the square inside the parentheses, we take half of the coefficient of \(x\), which is \(-4\), divide it by 2 to get \(-2\), and square it to get \(4\). Adding and subtracting \(4\) inside the parentheses gives us:
   \[
   3(x^2 - 4x + 4 - 4) = 3((x-2)^2 - 4)
   \]
3. Simplifying this expression, we get:
   \[
   3(x-2)^2 - 12
   \]
4. Now, rewrite the original expression including the constant term \(-11\):
   \[
   3(x-2)^2 - 12 - 11 = 3(x-2)^2 - 23
   \]

Therefore, the expression \(3x^2 - 12x - 11\) can be rewritten as \(3(x-2)^2 - 23\), which is in the form \(3(x + a)^2 + b\), where \(a = -2\) and \(b = -23\).

Would you like a more detailed explanation of any step? Here are some questions and a tip related to this topic:

1. Why do we factor out the 3 before completing the square?
2. How do you determine the term to add and subtract when completing the square?
3. Could you explain how completing the square works for any quadratic equation?
4. What happens if the quadratic coefficient (the coefficient of \(x^2\)) is not 1 when completing the square?
5. How can this method be used to solve quadratic equations?

**Tip:** When completing the square, always ensure you balance the equation by adding and subtracting the same value within the context of any factored constants.

The quadratic 3x^2 - 12x - 11 can be written in the form 3(x + a)^2 + b. Find the values of a and b.

Learn how to rewrite the quadratic equation 3x^2 - 12x - 11 in the form 3(x + a)^2 + b. This detailed step-by-step solution explains completing the square and is suitable for Grade 9-11 students.

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