Find the zeroes of the polynomial $4x^2 + 4x - 3$ and verify the relationship between zeroes and coefficients of the polynomial.
Generated by claude-sonnet-4-6 · 2026-06-15 10:34 · grounding rag
Model Answer
Finding zeroes:
$$4x^2 + 4x - 3 = 4x^2 + 6x - 2x - 3 = 2x(2x+3) - 1(2x+3) = (2x-1)(2x+3)$$
Zeroes: $2x - 1 = 0 \Rightarrow x = \dfrac{1}{2}$ and $2x + 3 = 0 \Rightarrow x = -\dfrac{3}{2}$
Verification (here $a = 4,\ b = 4,\ c = -3$):
$$\text{Sum of zeroes} = \frac{1}{2} + \left(-\frac{3}{2}\right) = -1 = \frac{-4}{4} = \frac{-b}{a} \checkmark$$
$$\text{Product of zeroes} = \frac{1}{2} \times \left(-\frac{3}{2}\right) = -\frac{3}{4} = \frac{-3}{4} = \frac{c}{a} \checkmark$$
Hence verified.
Source: Chapter 2, Section 2.3
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Explanation
- Splitting the middle term is the expected method: find two numbers whose product is $4 \times (-3) = -12$ and sum is $4$ → that's $6$ and $-2$.
- Always write the verification step using the formula $\alpha+\beta = -b/a$ and $\alpha\beta = c/a$, showing both sides equal. Examiners award 1 mark for finding zeroes and 2 marks for the full verification.
- Don't forget to identify $a$, $b$, $c$ before substituting — it avoids sign errors.