Determine whether the conjecture below is sometimes, always, or never true. Explain.
In a quadratic in standard form, if a and c are different signs, then the solutions will be real.
+2440 Let's consider the discriminant of a quadratic, \(b^2-4ac\).
If a and c are opposite signs, then ac will be negative. -4ac will be positive, then. b2 will also be positive--no matter whether b is positive or negative. This means that b2-4ac represents the addition of positive values, which will never be negative. Hence, the discriminant will always be positive, so the solutions can never be imaginary. This conjecture is always true.
+2440 Let's consider the discriminant of a quadratic, \(b^2-4ac\).
If a and c are opposite signs, then ac will be negative. -4ac will be positive, then. b2 will also be positive--no matter whether b is positive or negative. This means that b2-4ac represents the addition of positive values, which will never be negative. Hence, the discriminant will always be positive, so the solutions can never be imaginary. This conjecture is always true.
