State the relationship between zeroes and coefficients of a quadratic polynomial $ax^2 + bx + c$. Use this to find the zeroes of $f(x) = 4\sqrt{3} x^2 + 5x - 2\sqrt{3}$ and verify the relationship.
Step-by-Step Solution
Key Concept: Sum of zeroes $= -b/a$, Product of zeroes $= c/a$.<br>Split $5x = 8x - 3x$ since $4\sqrt{3} \times (-2\sqrt{3}) = -24$.
Statement: Sum $= -b/a$, Product $= c/a$. [1.0 Mark]
$4\sqrt{3} x^2 + 8x - 3x - 2\sqrt{3} = 0 \Rightarrow 4x(\sqrt{3} x + 2) - \sqrt{3}(\sqrt{3} x + 2) = 0$.
$(4x - \sqrt{3})(\sqrt{3} x + 2) = 0 \Rightarrow x = \sqrt{3}/4$ and $x = -2/\sqrt{3} = -2\sqrt{3}/3$. [2.0 Marks]
Verification:
Sum $= \sqrt{3}/4 - 2/\sqrt{3} = (3 - 8)/(4\sqrt{3}) = -5/(4\sqrt{3}) = -b/a$.
Product $= (\sqrt{3}/4)(-2/\sqrt{3}) = -2/4 = -2\sqrt{3}/(4\sqrt{3}) = c/a$. Verified! [2.0 Marks]
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🎯 Official CBSE Marking Scheme:
Stating sum $= -b/a$ and product $= c/a$: 1.0 Mark
Factoring and finding zeroes $\sqrt{3}/4$ and $-2/\sqrt{3}$: 2.0 Marks
Verifying sum and product relationships: 2.0 Marks
Correct Answer: