Question

При каких значениях x выражение -x^2+6x принимает отрицательные значения

Answer 1

Интервал значений х показан на фото.

Answer 2

To determine the values of x for which the expression -x^2 + 6x takes on negative values, we need to find the range of x for which the expression is less than zero.

Step 1: Set up the inequality: We want to find when -x^2 + 6x < 0.

Step 2: Factor the expression: -x^2 + 6x = x(-x + 6)

Step 3: Set each factor to zero and find critical points: x = 0 and -x + 6 = 0 Solving for x, we get: x = 0 and x = 6

Step 4: Create a number line and test each interval: Now, we can test the inequality in each of the following intervals created by the critical points: (-∞, 0), (0, 6), and (6, +∞).

Test for (-∞, 0): Pick a value less than 0, for example, x = -1: -(-1)^2 + 6(-1) = -1 - 6 = -7 (negative)

Test for (0, 6): Pick a value between 0 and 6, for example, x = 3: -(3)^2 + 6(3) = -9 + 18 = 9 (positive)

Test for (6, +∞): Pick a value greater than 6, for example, x = 7: -(7)^2 + 6(7) = -49 + 42 = -7 (negative)

Step 5: Interpret the results: The expression -x^2 + 6x takes negative values in the intervals (-∞, 0) and (6, +∞).

So, the values of x for which the expression -x^2 + 6x is negative are x < 0 and x > 6.