To rewrite the quadratic equation in the form [tex]\(3(x + a)^2 + b\)[/tex], we need to complete the square.
Given the quadratic equation:
[tex]\[ 3x^2 - 12x - 11 \][/tex]
### Step 1: Factor out the leading coefficient from the quadratic and linear terms
First, factor out 3 from the first two terms:
[tex]\[ 3(x^2 - 4x) - 11 \][/tex]
### Step 2: Complete the square inside the parenthesis
Next, complete the square inside the parenthesis. Recall how to complete the square for the expression [tex]\(x^2 - 4x\)[/tex]. We add and subtract the same value inside the parenthesis to create a perfect square trinomial.
[tex]\[ x^2 - 4x \][/tex]
Calculate the value needed to complete the square. To do this, take half of the linear coefficient (which is -4), square it, and then add and subtract it inside the parenthesis:
[tex]\[ \left( \frac{-4}{2} \right)^2 = (-2)^2 = 4 \][/tex]
So,
[tex]\[ x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4 \][/tex]
### Step 3: Substitute back and simplify
Substitute [tex]\( (x - 2)^2 - 4 \)[/tex] back into the equation:
[tex]\[ 3\left( (x - 2)^2 - 4 \right) - 11 \][/tex]
Distribute the 3:
[tex]\[ 3(x - 2)^2 - 3 \cdot 4 - 11 \][/tex]
[tex]\[ 3(x - 2)^2 - 12 - 11 \][/tex]
Combine the constants:
[tex]\[ 3(x - 2)^2 - 23 \][/tex]
Thus, we have the quadratic in the form [tex]\( 3(x + a)^2 + b \)[/tex]:
[tex]\[ 3(x + (-2))^2 - 23 \][/tex]
[tex]\[ 3(x - 2)^2 - 23 \][/tex]
### Final Answer
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = -2 \][/tex]
[tex]\[ b = -23 \][/tex]
