Multiple choice

If the roots of the equation $ax^2 + x + b = 0$ are real and different, then the roots of the equation $x^2 - 4 \sqrt{ab} x + 1 = 0$ will be

  1. Rational

  2. Irrational

  3. Real

  4. Imaginary
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For ax^2 + x + b = 0, real roots imply discriminant D = 1 - 4ab > 0, so 4ab < 1. For the second equation x^2 - 4*sqrt(ab)x + 1 = 0, the discriminant is D' = (4*sqrt(ab))^2 - 4(1)(1) = 16ab - 4. Since 4ab < 1, then 16ab < 4, which means 16ab - 4 < 0. A negative discriminant indicates imaginary roots.

AI explanation

Because the roots of ax squared plus x plus b are real and distinct, its discriminant is positive, meaning 1 minus 4ab is greater than 0 and ab is less than one quarter. The discriminant of the second quadratic, x squared minus 4 times the square root of ab times x plus 1, is calculated as 16ab minus 4. Since ab is less than one quarter, 16ab is less than 4, making the discriminant negative. A negative discriminant means the roots of the second equation are imaginary.

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