Final answer:
To solve the equation x² - 4kx - 32K² = 0 by factoring, find two numbers that multiply to -32K² and add up to -4k. Factor the equation accordingly and apply the zero product property to find the solutions for x.
Step-by-step explanation:
The equation given is x² - 4kx - 32K² = 0, which is a quadratic equation of the form ax² + bx + c = 0. To solve it by factoring, we need to find two numbers that multiply to give -32K² (the constant term) and add up to give -4k (the coefficient of x). Factoring is often a more straightforward method than using the quadratic formula when the equation can be easily decomposed into a product of binomials.
Let's assume the two numbers are m and n such that mn = -32K² and m + n = -4k. If we can find such numbers, the equation can be factored as (x + m)(x + n) = 0. We apply the zero product property, which states that if the product of two numbers is zero, then at least one of the numbers must be zero. Therefore, the solutions for x are the values that make each binomial equal to zero: x = -m and x = -n.
Example: If k = 2, the equation becomes x² - 8x - 128 = 0. Factoring, we might find that (x - 16)(x + 8) = 0, leading to the solutions x = 16 and x = -8.