Question

Решите неравенство (x^2-6x+5)(x+3)^2 <=0

Answer 1

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Answer 2

To solve the inequality $(x^2-6x+5)(x+3)^2 \leq 0$, we first find the critical points where the expression equals zero and then analyze the sign of the expression in the different intervals separated by these critical points.

Step 1: Find the critical points where the expression equals zero.

The expression is a polynomial, and to find the critical points, we set each factor equal to zero and solve for $x$:

1. $x^2 - 6x + 5 = 0$ This quadratic equation can be factored as follows: $(x - 5)(x - 1) = 0$

2. $x + 3 = 0$ Solving for $x$, we get $x = -3$

Step 2: Analyze the sign of the expression in different intervals.

Now, we have three critical points: $x = -3$, $x = 1$, and $x = 5$. These critical points divide the number line into four intervals:

Interval 1: $-\infty < x < -3$ Interval 2: $-3 < x < 1$ Interval 3: $1 < x < 5$ Interval 4: $x > 5$

Step 3: Test a point from each interval in the original inequality to determine the sign.

We can pick any value from each interval and evaluate the expression to determine its sign:

1. Test $x = -4$ (Interval 1): $((-4)^2 - 6(-4) + 5)(-4 + 3)^2 = (16 + 24 + 5)(-1)^2 = 45 \gt 0$ (positive)

2. Test $x = 0$ (Interval 2): $((0)^2 - 6(0) + 5)(0 + 3)^2 = (5)(3)^2 = 45 \gt 0$ (positive)

3. Test $x = 2$ (Interval 3): $((2)^2 - 6(2) + 5)(2 + 3)^2 = (4 - 12 + 5)(5)^2 = -3 \lt 0$ (negative)

4. Test $x = 6$ (Interval 4): $((6)^2 - 6(6) + 5)(6 + 3)^2 = (36 - 36 + 5)(9)^2 = 45 \gt 0$ (positive)

Step 4: Determine the solution based on the sign of the expression.

The inequality is satisfied when the expression is less than or equal to zero $(\leq 0)$, which means we need the expression to be negative.

From our tests, we see that the expression is negative in Interval 3 $(1 < x < 5)$. Therefore, the solution to the inequality is:

$1 < x < 5$

In interval notation, the solution is $(1, 5)$.