Final answer:
A quadratic equation ax² + bx + c = 0 can sometimes be factored into a perfect square trinomial if 'a' and 'c' are perfect squares. Otherwise, the quadratic formula, x = (-b ± √(b² - 4ac))/(2a), provides the roots of the equation regardless of the nature of 'a,' 'b,' and 'c.'
Step-by-step explanation:
The equation ax² + bx + c = 0 is a quadratic equation, which can be solved for x using various methods, one of which is factoring when certain conditions are met. If the coefficients a and c are perfect squares and b is twice the product of the square roots of a and c, the quadratic can be factored as a perfect square trinomial. Otherwise, the quadratic formula, which is x = (-b ± √(b² - 4ac))/(2a), can be used to find the roots of the equation regardless of whether a and c are perfect squares.
For equations that are not easily factored, the quadratic formula is a reliable and systematic way to find the solutions. To apply the formula, one simply needs to substitute the values of a, b, and c into the formula and perform the calculations. This method ensures that you find the solutions to the quadratic equation, whether they are real or complex numbers.