Given: $$3x^2 + 2x + 2 = 0$$ 2.1.1 Determine the numerical value of the discriminant (Δ) of the equation - NSC Technical Mathematics - Question 2 - 2020 - Paper 1

Since the discriminant (Δ) is negative ($Δ = -20$), it indicates that the quadratic equation has no real roots. Therefore, the roots are non-real (or complex) and appear as conjugate pairs.

Starting from the given equation: $x^2 - 2px = 3p^2$, we can rearrange it to the standard quadratic form: $x^2 - 2px - 3p^2 = 0$, where $a = 1$, $b = -2p$, and $c = -3p^2$.

To determine whether the roots are rational, we calculate the discriminant (Δ) for the equation $x^2 - 2px - 3p^2 = 0$:

Using the formula: $Δ = b^2 - 4ac$, we have: $Δ = (-2p)^2 - 4(1)(-3p^2)$ $Δ = 4p^2 + 12p^2$ $Δ = 16p^2$.

Since $16p^2$ is a perfect square (as $16$ and $p^2$ are both non-negative), it implies that the roots of the equation will also be rational.