If α and β are the roots of the equation 2x² - 3x - 5 = 0, then the quadratic equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) is:

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This question was previously asked in

RRB Group D 26 Sept 2022 Shift 2 Official Paper

Answer 1

Given:

The roots of equation 2x2 - 3x - 5 = 0 are α and β.

The roots of another equation are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

Calculation:

Using the hit and trial method on the given equation to find values of α and β.

2x2 - 3x - 5 = 0

= 2x2 - 5x + 2x - 5

= x(2x - 5) +1(2x - 5)

= (2x - 5)(x + 1)

x = \(\frac{5}{2}\) and x = -1

thus α = \(\frac{5}{2}\) and β = -1

Now solving, \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\), we get

\(\frac{\alpha}{\beta}\) = \(\frac{-5}{2}\)

\(\frac{\beta}{\alpha}\) = \(\frac{-2}{5}\)

Now,

Quadratic equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\):

(x - (\(\frac{-5}{2}\))) (x - (\(\frac{-2}{5}\))) = 0

⇒ (x + 5/2) (x + 2/5) = 0

⇒ x² + 5x/2 + 2x/5 + 1 = 0

⇒ (10x² + 25x + 4x + 10)/10 = 0

⇒ 10x² + 29x + 10 = 0

∴ The correct answer is option (2)

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