Question

2x²+5x-12>0 решите неравенство

Answer 1

$2x^2+5x-12\ \textgreater \ 0 \\ (x+4)(x- 1,5) \ \textgreater \ 0 \\ x \in (-\infty;-4)\cup( 1,5 ; +\infty)$

Answer 2

D=25-4*2*(-12)=25+96=121

x1=(-5+11)/2*2=6/4=1,5

x2=(-5-11)/2*2=-16/4=-4

Ответ: (-∞;-4)∪(1,5;+∞)

Answer 3

To solve the inequality 2x² + 5x - 12 > 0, we'll find the critical points and then determine the regions where the inequality is satisfied.

Step 1: Find the critical points: The critical points are the values of x where the expression 2x² + 5x - 12 equals zero. To find them, set the expression equal to zero and solve for x:

2x² + 5x - 12 = 0

This quadratic equation can be factored as follows:

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

Now, set each factor equal to zero and solve for x:

2x - 3 = 0 x = 3/2

x + 4 = 0 x = -4

So, the critical points are x = 3/2 and x = -4.

Step 2: Create a number line and determine the sign of the expression 2x² + 5x - 12 in each interval:

On the number line, we have three intervals defined by the critical points: (-∞, -4), (-4, 3/2), and (3/2, +∞).

Choose a test point from each interval and plug it into the expression 2x² + 5x - 12:

1. Test x = -5 (any value less than -4): 2(-5)² + 5(-5) - 12 = 50 - 25 - 12 = 13 (positive)

2. Test x = 0 (any value between -4 and 3/2): 2(0)² + 5(0) - 12 = 0 - 0 - 12 = -12 (negative)

3. Test x = 2 (any value greater than 3/2): 2(2)² + 5(2) - 12 = 8 + 10 - 12 = 6 (positive)

Step 3: Analyze the results and find the solution: The inequality is satisfied when the expression 2x² + 5x - 12 is greater than zero (positive).

From our test, we found that the inequality is satisfied in the intervals (-∞, -4) and (3/2, +∞). Therefore, the solution to the inequality is:

x < -4 or x > 3/2

In interval notation: (-∞, -4) U (3/2, +∞).