Question

X^2-2x-3>0 решить неравенство

Answer 1

X^2-2x-3>0
D=16
x=(2+-4)/2=3; -1
(x-3)(x+1)>0
парабола с ветвями вверх,
будет значение >0 в промежутках вне корней
x принадлежит (-oo;-1)u(3;+oo)

Answer 2

To solve the inequality $x^2 - 2x - 3 > 0$, we can use various methods, such as graphing, factoring, or using the quadratic formula. Let's use factoring in this case:

Step 1: Factor the quadratic expression $x^2 - 2x - 3$:

$x^2 - 2x - 3 = (x - 3)(x + 1)$

Step 2: Set each factor to zero to find the critical points:

$x - 3 = 0 \Rightarrow x = 3$

$x + 1 = 0 \Rightarrow x = -1$

Step 3: Now, we have three intervals on the number line to test for the inequality:

• Interval 1: $x < -1$
• Interval 2: $-1 < x < 3$
• Interval 3: $x > 3$

Step 4: Test each interval with a test point (any value within that interval) to determine the sign of $x^2 - 2x - 3$:

• For $x < -1$, let's pick $x = -2$: $f(-2) = (-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 0$

• For $-1 < x < 3$, let's pick $x = 0$: $f(0) = (0 - 3)(0 + 1) = (-3)(1) = -3 < 0$

• For $x > 3$, let's pick $x = 4$: $f(4) = (4 - 3)(4 + 1) = (1)(5) = 5 > 0$

Step 5: Analyze the results:

The inequality $x^2 - 2x - 3 > 0$ is satisfied when $x < -1$ or $x > 3$ because in those intervals, $x^2 - 2x - 3$ is positive. However, it is not satisfied when $-1 < x < 3$ because in that interval, $x^2 - 2x - 3$ is negative.

So, the solution to the inequality is $x < -1$ or $x > 3$.