Multiple choice

If the coefficient of $x^2$ and the constant term have the same sign and if the coefficient of $x$ term is zero, then the quadratic equation has :

  1. no real roots

  2. one real root

  3. one real and one imaginary roots

  4. two real roots
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A Correct answer
Explanation

If the x term is zero, the equation is ax^2 + c = 0. If a and c have the same sign, then ax^2 = -c, so x^2 = -c/a. Since c and a have the same sign, -c/a is negative. The square of a real number cannot be negative, so there are no real roots.

AI explanation

For a quadratic equation ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac. If the coefficient of x (b) is zero, the discriminant simplifies to D = -4ac. Since the coefficient of x^2 (a) and the constant term (c) have the same sign, their product ac is positive, which makes the discriminant -4ac a negative number. A negative discriminant means the equation has no real roots.

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