Which of the following equations has no real roots?

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bb(x^2 - 4x + 3sqrt(2) = 0) Explanation: (A) The given equation is x^2 - 4x + 3sqrt(2) = 0 On comparing with ax2 + bx + c = 0, we get a = 1, b = – 4 and c = 3sqrt(2) The discriminant of x^2 - 4x + 3sqrt(2) = 0 is D = b2 – 4ac = (-4)^2 - 4(1)(3sqrt(2)) = 16 - 12sqrt(2) = 16 – 12 $\times$ (1.41) = 16 – 16.92 = – 0.92 $\Rightarrow$ b2 – 4ac < 0 (B) The given equation is x^2 + 4x - 3sqrt(2) = 0 On comparing the equation with ax2 + bx + c = 0, we get a = 1, b = 4 and c = -3sqrt(2) Then, D = b2 – 4ac = (-4)^2 - 4(1)(-3sqrt(2)) = 16 + 12sqrt(2) > 0 Hence, the equation has real roots. (C) Given equation is x^2 - 4x - 3sqrt(2) = 0 On comparing the equation with ax2 + bx + c = 0, we get a = 1, b = – 4 and c = -3sqrt(2) Then, D = b2 – 4ac = (-4)^2 - 4(1)(-3sqrt(2)) = 16 + 12sqrt(2) > 0 Hence, the equation has real roots. (D) Given equation is 3x^2 + 4sqrt(3)x + 4 = 0 On comparing the equation with ax2 + bx + c = 0, we get a = 3, b = 4sqrt(3) and c = 4 Then, D = b2 – 4ac = (4sqrt(3))^2 - 4(3)(4) = 48 – 48 = 0 Hence, the equation has real roots. Hence, x^2 - 4x + 3sqrt(2) = 0 has no real roots.

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