Multiple choice

The roots of the equation $a^2x^2 + (a + b)x - b^2 = 0$ are

  1. real and different

  2. real and equal

  3. imaginary

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The discriminant D = (a + b)^2 - 4(a^2)(-b^2) = a^2 + 2ab + b^2 + 4a^2b^2. Since a^2, b^2, and 4a^2b^2 are non-negative, D is positive for any non-zero real a and b. Thus, the roots are real and different.

AI explanation

The discriminant determines the nature of the roots, calculated as D equals b squared minus 4ac. For the given equation, the coefficients are a equal to a squared, b equal to a plus b, and c equal to negative b squared, so D equals the quantity a plus b squared minus four times a squared times negative b squared. This simplifies to a squared plus two ab plus b squared plus four a squared b squared, which is the sum of squares and thus strictly positive for real values. A positive discriminant means the roots are real and different.

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