Let P(x) = -x4 + 3x² + 5. Does the following equation have a root in the interval [2, 3]? -x4 + 3x² + 5 = 0

Question

Let P(x) = -x⁴ + 3x² + 5.

Does the following equation have a root in the interval [2, 3]?

-x⁴ + 3x² + 5 = 0

Is it A, B, C, or D?

A. Yes, because P(2) > 0 and P(3) < 0		B. Not necessarily, because P(2) > 0 and P(3) > 0
	C. Yes, because P(2) < 0 and P(3) > 0		D. Not necessarily, because P(2) < 0 and P(3) < 0

Answer 1

Looking at this graphically is the best approach. A quadratic equation where the coefficient of the X^2 term is positive opens upwards. In that configuration, if P(2) and P(3) are both positive, then we are not sure that the graph dips below the y axis in the interval. So the answer is B. Testing the discriminant (b^2-4ac) verifies the answer - the discriminant is negative, so there are no real roots.