Question

(2x+5)^2>=(5x-2)^2 решите пожалуйста в ответе должно получится -3/7 и 7/3 а у меня не получается

Answer 1

на фото...................

Answer 2

Let's solve the inequality step by step to find the correct solution. We'll start by simplifying both sides of the inequality:

(2x + 5)^2 ≥ (5x - 2)^2

Expanding both sides:

(2x + 5)(2x + 5) ≥ (5x - 2)(5x - 2)

Now, apply the FOIL method (First, Outer, Inner, Last) to expand both sides:

4x^2 + 20x + 25 ≥ 25x^2 - 20x + 4

Now, bring all terms to one side of the inequality:

4x^2 + 20x + 25 - 25x^2 + 20x - 4 ≥ 0

Simplify:

-21x^2 + 40x + 21 ≥ 0

To solve this quadratic inequality, we need to find the critical points where the inequality may change direction. The critical points are the solutions to the equation -21x^2 + 40x + 21 = 0.

To find the critical points, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

In our case, a = -21, b = 40, and c = 21:

x = [ -40 ± √(40^2 - 4(-21)(21)) ] / 2(-21) x = [ -40 ± √(1600 + 1764) ] / (-42) x = [ -40 ± √3364 ] / (-42) x = [ -40 ± 58 ] / (-42)

Now, we get two potential critical points:

1. x = (-40 + 58) / (-42) = 18 / (-42) = -3 / 7
2. x = (-40 - 58) / (-42) = -98 / (-42) = 7 / 3

Now, let's test the intervals created by these critical points. We can use a test point from each interval to determine the sign of -21x^2 + 40x + 21.

Test x = 0 (pick a value less than -3/7, e.g., x = -1):

-21(-1)^2 + 40(-1) + 21 = -21 + 40 + 21 = 40 > 0 (positive)

Test x = 1 (pick a value between -3/7 and 7/3):

-21(1)^2 + 40(1) + 21 = -21 + 40 + 21 = 40 > 0 (positive)

Test x = 2 (pick a value greater than 7/3, e.g., x = 3):

-21(3)^2 + 40(3) + 21 = -189 + 120 + 21 = -48 < 0 (negative)

The signs of -21x^2 + 40x + 21 in the intervals are:

• Negative for x < -3/7
• Positive for -3/7 < x < 7/3
• Negative for x > 7/3

We want to find where the inequality -21x^2 + 40x + 21 ≥ 0 holds. Thus, we are interested in the intervals where the expression is non-negative.

The solution is: -3/7 ≤ x ≤ 7/3.

So, your initial claim is incorrect. The correct solution to the inequality is -3/7 ≤ x ≤ 7/3.