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Relation between Roots and Coefficients for Quadratic

If α, β are t...

Question

If $α, β$ are the roots of $2 x^{2} - 3 x + 4 = 0$ , then the equation whose roots are $α^{2}$ and $β^{2}$ is

A

4 x^{2} + 7 x + 16 = 0

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B

4 x^{2} - 7 x + 16 = 0

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C

4 x^{2} + 7 x - 16 = 0

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D

4 x^{2} - 7 x - 16 = 0

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Solution

The correct option is A $4 x^{2} + 7 x + 16 = 0$
Let $f (x) = 2 x^{2} - 3 x + 4 = 0 \to$ roots are $α, β$
The equation whose roots are $α^{2}, β^{2}$ is $f (\sqrt{x}) = 0$
$\Rightarrow 2 x - 3 \sqrt{x} + 4 = 0 \Rightarrow 2 x + 4 = 3 \sqrt{x}$
$\Rightarrow (2 x + 4)^{2} = 9 x \Rightarrow 4 x^{2} + 7 x + 16 = 0$

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