Question

-2X^2-2X+6≥0 СРОЧНО

Answer 1

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Answer 2

To solve the inequality -2X^2 - 2X + 6 ≥ 0, we need to find the values of X that make the expression greater than or equal to zero.

Step 1: First, let's find the critical points by setting the expression equal to zero and solving for X: -2X^2 - 2X + 6 = 0

Step 2: To solve this quadratic equation, we can use the quadratic formula: X = (-b ± √(b^2 - 4ac)) / 2a

Where a = -2, b = -2, and c = 6.

X = (2 ± √((-2)^2 - 4 * (-2) * 6)) / (2 * (-2)) X = (2 ± √(4 + 48)) / (-4) X = (2 ± √52) / (-4) X = (2 ± 2√13) / (-4) X = (1 ± √13) / (-2)

So the critical points are X = (1 + √13) / (-2) and X = (1 - √13) / (-2).

Step 3: Now, we have three intervals to check: (-∞, (1 - √13) / (-2)), ((1 - √13) / (-2), (1 + √13) / (-2)), and ((1 + √13) / (-2), ∞).

Step 4: Test a value within each interval to determine the sign of the expression -2X^2 - 2X + 6:

1. Test X = 0: -2(0)^2 - 2(0) + 6 = 6 ≥ 0 (positive)

2. Test X = -1: -2(-1)^2 - 2(-1) + 6 = 6 ≥ 0 (positive)

3. Test X = -3: -2(-3)^2 - 2(-3) + 6 = -12 - (-6) + 6 = -12 < 0 (negative)

Step 5: Now, we can determine the sign of the expression for each interval:

(-∞, (1 - √13) / (-2)): The expression is positive. ((1 - √13) / (-2), (1 + √13) / (-2)): The expression is positive. ((1 + √13) / (-2), ∞): The expression is negative.

Step 6: Finally, we are looking for the values of X for which the expression is greater than or equal to zero:

The solution to the inequality -2X^2 - 2X + 6 ≥ 0 is: X ∈ ((1 - √13) / (-2), (1 + √13) / (-2)].