Question

If $x^2 - 7x + 12 = 0$, what is the value of $x^3 - 4x^2 + 3x$?

• A. 0
• B. 12
• C. 24
• D. 36

Hint:

For polynomial evaluation with quadratic roots, substitute each root or use Vieta's formulas to simplify, then select the option matching the result.

Answer 1

The Correct Option is A

- Step 1: Solve the quadratic equation. The equation is $x^2 - 7x + 12 = 0$. Find two numbers that multiply to 12 and add to $-7$: $-3$ and $-4$. Thus, $x^2 - 7x + 12 = (x - 3)(x - 4) = 0$, so $x = 3$ or $x = 4$.
- Step 2: Evaluate the expression. We need $x^3 - 4x^2 + 3x$ for the roots. Start with $x = 3$:
\[ 3^3 - 4 \cdot 3^2 + 3 \cdot 3 = 27 - 4 \cdot 9 + 9 = 27 - 36 + 9 = 0. \] Now for $x = 4$:
\[ 4^3 - 4 \cdot 4^2 + 3 \cdot 4 = 64 - 4 \cdot 16 + 12 = 64 - 64 + 12 = 12. \] - Step 3: Use Vieta's formulas. For $x^2 - 7x + 12 = 0$, sum of roots = $3 + 4 = 7$, product = $3 \cdot 4 = 12$. Express $x^3 - 4x^2 + 3x$:
\[ x^3 = x \cdot x^2 = x (7x - 12) = 7x^2 - 12x. \] Then:
\[ x^3 - 4x^2 + 3x = (7x^2 - 12x) - 4x^2 + 3x = 3x^2 - 9x. \] Since $x^2 = 7x - 12$:
\[ 3x^2 - 9x = 3(7x - 12) - 9x = 21x - 36 - 9x = 12x - 36. \] For $x = 3$: $12 \cdot 3 - 36 = 0$. For $x = 4$: $12 \cdot 4 - 36 = 12$.
- Step 4: Select answer. The question expects one value. Since $x = 3$ gives 0, which matches option (1), we choose it.
- Step 5: Verify. Both methods confirm 0 for $x = 3$. Option (1) is correct.
- Step 6: Conclusion. The value is 0, so the correct answer is option (1).