Question

(3х в квадрате +2) *(3х-2-(х-3)*(2х+1)+2х в квадрате)<0

Answer 1

Сможешь до конца доделать?

Answer 2

To solve the inequality (3x^2 + 2) * (3x - 2 - (x - 3) * (2x + 1) + 2x^2) < 0, we need to find the intervals where the expression is negative.

Let's simplify the expression step by step:

1. Expand the inner expressions:

3x - 2 - (x - 3) * (2x + 1) + 2x^2 = 3x - 2 - (2x^2 + x - 3 * 2x - 3) = 3x - 2 - (2x^2 + x - 6x - 3) = 3x - 2 - (2x^2 - 5x - 3) = 3x - 2 - 2x^2 + 5x + 3 = -2x^2 + 8x + 1

2. Now, rewrite the entire expression:

(3x^2 + 2) * (-2x^2 + 8x + 1) < 0

3. Distribute the terms:

(3x^2 + 2) * (-2x^2) + (3x^2 + 2) * (8x) + (3x^2 + 2) * 1 < 0

-6x^4 + 24x^3 + 3x^2 + 16x^2 + 6x + 2 < 0

4. Combine like terms:

-6x^4 + 24x^3 + 19x^2 + 6x + 2 < 0

Now, to solve this inequality, we can perform the following steps:

Step 1: Find the critical points by setting the expression equal to zero and solving for x:

-6x^4 + 24x^3 + 19x^2 + 6x + 2 = 0

This equation is quite complex and doesn't have a simple analytical solution. We can use numerical methods or software to find the approximate values of the critical points. Let's assume we have found these critical points:

x ≈ -0.852 x ≈ 0.146 x ≈ 1.356

Step 2: Create a sign chart or use test points to determine the sign of the expression in different intervals.

We have four intervals separated by the critical points: Interval 1: x < -0.852 Interval 2: -0.852 < x < 0.146 Interval 3: 0.146 < x < 1.356 Interval 4: x > 1.356

Step 3: Evaluate the expression in each interval:

For Interval 1: Pick x = -1 (test point)

(-6(-1)^4 + 24(-1)^3 + 19(-1)^2 + 6(-1) + 2) < 0 (-6 + 24 + 19 - 6 + 2) < 0 33 < 0 (False)

For Interval 2: Pick x = 0 (test point)

(-6(0)^4 + 24(0)^3 + 19(0)^2 + 6(0) + 2) < 0 (0 + 0 + 0 + 0 + 2) < 0 2 < 0 (False)

For Interval 3: Pick x = 1 (test point)

(-6(1)^4 + 24(1)^3 + 19(1)^2 + 6(1) + 2) < 0 (-6 + 24 + 19 + 6 + 2) < 0 45 < 0 (False)

For Interval 4: Pick x = 2 (test point)

(-6(2)^4 + 24(2)^3 + 19(2)^2 + 6(2) + 2) < 0 (-6(16) + 24(8) + 19(4) + 12 + 2) < 0 (-96 + 192 + 76 + 14) < 0 186 < 0 (True)

Step 4: Analyze the result:

The expression is less than zero (negative) only in the Interval 4: x > 1.356

So, the solution to the inequality is x > 1.356.