Let x1 < x2 < x3 be three real roots of equation \sqrt{2014} \textrm{ }\textrm{ } x^{3} – \textrm{ }\text{4} 0 \left(\text{29x}\right)^{\text{2}} + \textrm{ }\text{2}\textrm{ } = \textrm{ } 0 . Find the value of x2(x1 + x3).

Engineering

Mathematics

Quadratic Equation and Nature of Roots

Question

Let x₁ < x₂ < x₃ be three real roots of equation $\sqrt{2014} x^{3} - 40 {29x}^{2} + 2 = 0 .$

Find the value of x₂(x₁ + x₃).

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$Let \sqrt{2014} = a$

Given equation becomes,

2x²a² – x³a + x² – 2 = 0 [4029 = 2014(2) + 1]

$a = \frac{x^{3} \pm (x^{3} - 4 x)}{4 x^{2}}$

$∴ a = \frac{1}{x} or a = \frac{x^{2} - 2}{2 x} .$