Question

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

Hint:

For quadratic polynomials \( ax^2 + bx + c \), sum of roots \( = -b/a \), product of roots \( = c/a \).

Answer 1

The given quadratic equation is:
\[ x^2 - 3 = 0. \] Solving for \( x \):
\[ x = \pm \sqrt{3}. \] Now, sum of the roots:
\[ \alpha + \beta = \sqrt{3} + (-\sqrt{3}) = 0. \] Product of the roots:
\[ \alpha \beta = (\sqrt{3}) \times (-\sqrt{3}) = -3. \] Comparing with the standard quadratic form \( ax^2 + bx + c = 0 \):
\[ \text{Sum} = -\frac{b}{a} = 0, \quad \text{Product} = \frac{c}{a} = -3. \] The relationship holds.