Quadratic equations are polynomial equations of degree 2, written in the form ax² + bx + c = 0, where a, b, and c are constants. The general form of a quadratic equation allows us to determine its roots or solutions, which are the values of x that satisfy the equation.

In this case, we are looking for a quadratic equation with roots that are equal in magnitude but opposite in sign. This means that if one root is "r", the other root would be "-r".

To find such an equation, we need to consider the properties of the roots of a quadratic equation. The sum of the roots of a quadratic equation is given by the formula -b/a, and the product of the roots is given by the formula c/a.

Let's break down the options and analyze them:

a. ax² + bx + c = 0, a, b, c > 0
This option does not necessarily guarantee roots that are equal in magnitude and opposite in sign. The coefficients a, b, and c being positive does not provide any specific conditions on the roots of the equation.

b. ax² + bx = 0, a, b > 0
In this case, the coefficient c is not present in the equation. Therefore, the product of the roots would be c/a = 0. This implies that at least one of the roots is zero. However, having roots equal in magnitude and opposite in sign is not possible with this equation.

c. ax² + c = 0, a > 0, c > 0
This option, where the coefficient b is not present in the equation, offers a possibility for roots that are equal in magnitude and opposite in sign. Let's analyze it further:

- The sum of the roots would be -b/a = 0, as b is not present.
- The product of the roots would be c/a, which is positive since both c and a are positive.

From the conditions above, we can conclude that the roots of this equation are equal in magnitude (0) but opposite in sign, satisfying the given criteria.

Therefore, the correct answer is option c. ax² + c = 0, a > 0, c > 0.