Multiple choice

The roots of the equation 4x2 – 2(a2 + b2)x + a2b2 = 0 are

  1. a2/2, b2/2

  2. ab, a2b2

  3. a2, b2

  4. a2/b, b2/a

  5. Cannot be Determined
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A Correct answer
Explanation

The equation is 4x^2 - 2(a^2 + b^2)x + a^2b^2 = 0. Using the quadratic formula or factoring, the roots are (a^2)/2 and (b^2)/2.

AI explanation

We compare the given quadratic equation 4x^2 - 2(a^2 + b^2)x + a^2b^2 = 0 with the standard form Ax^2 + Bx + C = 0 to get A = 4, B = -2(a^2 + b^2), and C = a^2b^2. Using the quadratic formula, x = [-B plus or minus the square root of (B^2 - 4AC)] divided by 2A. The discriminant B^2 - 4AC equals 4(a^2 + b^2)^2 - 16(a^2b^2), which simplifies to 4a^4 - 8a^2b^2 + 4b^4, or 4(a^2 - b^2)^2. The square root of the discriminant is 2(a^2 - b^2), so applying the formula gives x = [2(a^2 + b^2) plus or minus 2(a^2 - b^2)] divided by 8. Solving this yields the two roots as a^2/2 and b^2/2.

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