Question

Find the zeroes of the quadratic polynomial \( 3x^2 - x - 4 \) and verify the relationship between the zeroes and the coefficients.

Hint:

Zeroes of Quadratic Equation:Use the quadratic formula and verify sum-product properties.

Answer 1

Using the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-4)}}{2(3)} \] \[ x = \frac{1 \pm \sqrt{1 + 48}}{6} \] \[ x = \frac{1 \pm \sqrt{49}}{6} \] \[ x = \frac{1 \pm 7}{6} \] \[ x = \frac{8}{6} = \frac{4}{3}, \quad x = \frac{-6}{6} = -1 \] Verifying:
Sum of zeroes:
\[ \frac{4}{3} + (-1) = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} = -\frac{b}{a} = -\frac{-1}{3} \] Product of zeroes:
\[ \frac{4}{3} \times (-1) = -\frac{4}{3} = \frac{c}{a} = \frac{-4}{3} \] Thus, the relationship is verified.
Correct Answer: \( \frac{4}{3}, -1 \) are the zeroes, relationship verified.