Final answer:
One root of the quadratic equation ax^2 + bx + c = 0 is a positive real number when a > 0 and c < 0 because the product of the roots must be negative, suggesting opposite signs for the roots. The use of the quadratic formula further confirms that the roots are real and one of them will indeed be positive. Therefore, the statement is true.
Step-by-step explanation:
To prove whether one solution of the quadratic equation ax^2 + bx + c = 0 is a positive real number when a > 0 and c < 0, we can rely on the fact that the product of the roots is equal to c/a, according to Viète's formulas. Because a > 0 and c < 0, the product of the roots would be negative, which implies that the roots must have opposite signs, one being positive and the other being negative. Therefore, one solution of the quadratic equation is indeed a positive real number.
The quadratic formula, which is given by x = −b ± √(b^2 - 4ac) / (2a), determines the exact solutions for x that satisfy the quadratic equation. Considering that a, b, and c are real numbers, with a > 0 and c < 0, the discriminant b^2 - 4ac becomes positive, and since you're dividing by 2a (which is positive), you'll get two real roots, one positive and one negative. Consequently, the statement in question is True.