Final answer:
The equation x^4 - 3x^2 - 28 = 0 has real solutions x = ±√7 and imaginary solutions x = ±2i.
Step-by-step explanation:
The equation x^4 - 3x^2 - 28 = 0 can be solved by factoring and then using the quadratic formula. Let's solve it step by step:
Step 1: Factor the equation as (x^2 - 7)(x^2 + 4) = 0.
Step 2: Set each factor equal to zero and solve the resulting quadratic equations.
First factor: x^2 - 7 = 0. Solving this, we get x = ±√7.
Second factor: x^2 + 4 = 0. This quadratic equation has no real solutions because it involves a squared term with a positive coefficient. However, it does have imaginary solutions. Using the quadratic formula, we get x = ±2i, where i is the imaginary unit.
Therefore, the real solutions are x = ±√7 and the imaginary solutions are x = ±2i.