Question

(x-5)2 (x+3) (x-4)>=0

Answer 1

=2х-10(х+3)(х-4)=2х-10х-30(х-4)=2х-10х-30х+120=-38х+120

Answer 2

To solve the inequality (x-5)^2 * (x+3) * (x-4) >= 0, you can use the following steps:

1. Find the critical points of the inequality by setting each factor equal to zero and solving for x:

   a) (x-5)^2 = 0 (x-5)(x-5) = 0 x = 5

   b) (x+3) = 0 x = -3

   c) (x-4) = 0 x = 4

2. Plot these critical points on a number line:

   -3 | 4 | 5

3. Test the intervals between and around the critical points. Choose test points in each interval and evaluate the expression (x-5)^2 * (x+3) * (x-4) at these points.

   Test point in (-∞, -3): x = -4 ((-4 - 5)^2) * (-4 + 3) * (-4 - 4) = (9) * (-1) * (-8) = 72

   Test point in (-3, 4): x = 0 ((0 - 5)^2) * (0 + 3) * (0 - 4) = (25) * (3) * (-4) = -300

   Test point in (4, 5): x = 4.5 ((4.5 - 5)^2) * (4.5 + 3) * (4.5 - 4) = (0.25) * (7.5) * (0.5) = 0.9375

   Test point in (5, ∞): x = 6 ((6 - 5)^2) * (6 + 3) * (6 - 4) = (1) * (9) * (2) = 18

4. Based on the test results, you can determine the solution to the inequality:

   -3 | 4 | 5

   • • +

The sign of the expression (x-5)^2 * (x+3) * (x-4) is positive in the intervals (-∞, -3) and (5, ∞) because the test points in these intervals yielded positive values, while it is negative in the interval (-3, 4) because the test point in this interval yielded a negative value.

So, the solution to the inequality (x-5)^2 * (x+3) * (x-4) >= 0 is:

x in (-∞, -3] U [5, ∞)