Final answer:
To solve the inequality f(x) ≤ 0 for the given function, find the roots using the quadratic formula. The roots are x = 3 and x = -8. The function is less than or equal to zero between these roots, so the solution is -8 ≤ x ≤ 3.
Step-by-step explanation:
To solve the inequality f(x) ≤ 0 for the function f(x) = x²+5x-24, we first identify the values of x for which f(x) is less than or equal to zero. This involves finding the roots of the quadratic equation when f(x) = 0. We can use the quadratic formula, x = (-b ± √(b²-4ac))/(2a), where a, b, and c are coefficients from the quadratic equation ax²+bx+c=0.
Applying the quadratic formula here, we find the roots to be:
x = (-5 ± √(5²-4(1)(-24)))/(2(1))
x = (-5 ± √(25+96))/2
x = (-5 ± √121)/2
x = (-5 ± 11)/2
So the roots are x = 3 and x = -8.
Since we are looking for the values of x where f(x) is less than or equal to zero, we test intervals around the roots and find that the function is negative between x = -8 and x = 3.
The solution to the inequality is therefore -8 ≤ x ≤ 3.