Question

Give an example of a quadratic function that has no real root but has complex solutions and solve for them algebraically.

Answer 1

The quadratic function f(x) = x² + 4 has no real roots, but it has complex solutions which can be found using the quadratic formula. These solutions are x = 2i and x = -2i.

Step-by-step explanation:

An example of a quadratic function that has no real root but has complex solutions can be f(x) = x² + 4. To solve for the roots algebraically, we use the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a) where a, b, and c are coefficients of the equation ax² + bx + c = 0. In this case, a = 1, b = 0, and c = 4. Plugging these into the formula, we get:

x = (-0 ± √(0² - 4(1)(4))) / (2(1)),
x = (± √(-16)) / 2,
x = ± 4i / 2,
x = ± 2i.

Therefore, the complex solutions to this quadratic equation are x = 2i and x = -2i.