Multiple choice

If the coefficient of $x^2$ and the constant term of a quadratic equation have opposite signs, then the quadratic equation has _______ roots.

  1. Real and Distinct Roots

  2. Real and Equal Roots

  3. Imaginary Roots

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In ax^2 + bx + c = 0, the discriminant is D = b^2 - 4ac. If a and c have opposite signs, then ac < 0, so -4ac > 0. Since b^2 >= 0, D = b^2 - 4ac > 0, ensuring real and distinct roots.

AI explanation

In a standard quadratic equation ax^2 + bx + c = 0, the product of the roots is given by c/a. If the coefficient of x^2 (a) and the constant term (c) have opposite signs, their product ac is a negative number. Because the discriminant is D = b^2 - 4ac, and ac is negative, -4ac becomes a positive value, ensuring that b^2 - 4ac is always greater than zero. Therefore, the equation will always have real and distinct roots.

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