3x^2-11x^2+16x-12=0 when one of the roots is x=2

Let's solve the quadratic equation \( 3x^2 - 11x^2 + 16x - 12 = 0 \), given that one of the roots is \( x = 2 \).

### Step 1: Simplify the equation
Combine the like terms for \( x^2 \):
\[
3x^2 - 11x^2 = -8x^2
\]
So the equation becomes:
\[
-8x^2 + 16x - 12 = 0
\]
We can simplify further by multiplying through by \( -1 \) to make the leading coefficient positive:
\[
8x^2 - 16x + 12 = 0
\]

### Step 2: Use the fact that \( x = 2 \) is a root
Since \( x = 2 \) is a root, substitute \( x = 2 \) into the equation:
\[
8(2)^2 - 16(2) + 12 = 0
\]
\[
8(4) - 32 + 12 = 0
\]
\[
32 - 32 + 12 = 0
\]
\[
12 \neq 0
\]
This indicates an inconsistency. Therefore, the assumption that \( x = 2 \) is a root does not satisfy the equation \( 8x^2 - 16x + 12 = 0 \).

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3x^2 - 11x^2 + 16x - 12 = 0 when one of the roots is x = 2

Solve the quadratic equation 3x^2 - 11x^2 + 16x - 12 with one of the roots being x = 2. This problem involves simplifying quadratic equations, using algebraic concepts, and applying the factor theorem. Suitable for students in Grades 9-10.

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