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Question: If α and β are roots of the equation ax<sup>2</sup> + bx + c = 0 then roots of the equation a(2x + 1...

If α and β are roots of the equation ax2 + bx + c = 0 then roots of the equation a(2x + 1)2 – b(2x + 1)(3 – x) + c(3 – x)2 = 0

are:

A

2α+1α–3\frac{2\alpha + 1}{\alpha –3},2β+1β–3\frac{2\beta + 1}{\beta –3}

B

3α+1α–2\frac{3\alpha + 1}{\alpha –2},3β+1β–2\frac{3\beta + 1}{\beta –2}

C

2α–1α–2\frac{2\alpha –1}{\alpha –2},2β+1β–2\frac{2\beta + 1}{\beta –2}

D

None of these

Answer

3α+1α–2\frac{3\alpha + 1}{\alpha –2},3β+1β–2\frac{3\beta + 1}{\beta –2}

Explanation

Solution

a(2x+1)2(x–3)2\frac{(2x + 1)^{2}}{(x–3)^{2}} + b(2x+1)(x–3)\frac{(2x + 1)}{(x–3)} + c = 0

⇒ 2x+1x–3\frac{2x + 1}{x–3} = α or 2x+1x–3\frac{2x + 1}{x–3} = β

⇒ 2x + 1 = αx – 3α

⇒ x (α – 2) = 1 + 3α

⇒ x = 1+3αα–2\frac{1 + 3\alpha}{\alpha –2}, 1+3ββ–2\frac{1 + 3\beta}{\beta –2}