Multiple choice

State the nature of the given quadratic equation $\sqrt{2}x^2-\cfrac{3}{\sqrt{2}}x+\cfrac{1}{\sqrt{2}}= 0$

  1. Real and Distinct roots

  2. Real and equal roots

  3. Imaginary roots

  4. None of the Above
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A Correct answer
Explanation

The discriminant D = b^2 - 4ac. Here, a = sqrt(2), b = -3/sqrt(2), c = 1/sqrt(2). D = (-3/sqrt(2))^2 - 4(sqrt(2))(1/sqrt(2)) = 9/2 - 4 = 4.5 - 4 = 0.5. Since D > 0, the roots are real and distinct.

AI explanation

To find the nature of the roots, evaluate the discriminant using the formula D = b^2 - 4ac. For the equation (2^(1/2))x^2 - 3/(2^(1/2))x + 1/(2^(1/2)) = 0, the coefficients are a = 2^(1/2), b = -3/(2^(1/2)), and c = 1/(2^(1/2)). Calculating D gives (-3/(2^(1/2)))^2 - 4(2^(1/2))(1/(2^(1/2))), which simplifies to 9/2 - 4 = 1/2. Since 1/2 is greater than zero, the equation has real and distinct roots.

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