Find the zeroes of the polynomial $f(x) = 4\sqrt{3}x^2 + 5x - 2\sqrt{3}$.
Step-by-Step Solution
Key Concept: Split the middle term: product $= (4\sqrt{3})(-2\sqrt{3}) = -24$, sum $= 5$. The factors are $8$ and $-3$.
$f(x) = 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3} = 4x(\sqrt{3}x + 2) - \sqrt{3}(\sqrt{3}x + 2) = (4x - \sqrt{3})(\sqrt{3}x + 2)$. [1.0 Mark]
Setting $f(x) = 0 \Rightarrow 4x - \sqrt{3} = 0$ or $\sqrt{3}x + 2 = 0 \Rightarrow x = \dfrac{\sqrt{3}}{4}$ or $x = -\dfrac{2}{\sqrt{3}} = -\dfrac{2\sqrt{3}}{3}$. Zeroes are $\dfrac{\sqrt{3}}{4}$ and $-\dfrac{2}{\sqrt{3}}$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Correct middle term split and factorisation: 1.0 Mark
Finding both zeroes correctly: 1.0 Mark
Correct Answer: