Question

X^2-3x-10 меньше или равно 0

Answer 1

Ответ:

x ∈ [-2; 5]

Пошаговое объяснение:

D = 9 + 40 = 49

x₁ = (3 - 7)/2 = -2

x₂ = (3 + 7)/2 = 5

___[-2]/_/_/_/_/_/_/[5]_____

Answer 2

Д=9-4*1*(-10)=49

х1=(3+7)/2=10/2=5

х2=(3-7)/2=-4/2=-2

_____-2__________5_____>

...........//////////////

[-2; 5]

Answer 3

To find the solution to the inequality $x^2 - 3x - 10 \leq 0$, we need to determine the values of $x$ that satisfy this inequality.

Step 1: Factorize the quadratic expression: $x^2 - 3x - 10 = (x - 5)(x + 2)$

Step 2: Set each factor to zero and find critical points: $x - 5 = 0 \Rightarrow x = 5$ $x + 2 = 0 \Rightarrow x = -2$

Step 3: Plot these critical points on a number line:

$-\infty \quad -2 \quad 5 \quad \infty$

Step 4: Test points in each interval to determine the sign of $x^2 - 3x - 10$:

• Test $x = -3$ (pick any value between $-\infty$ and $-2$): $((-3) - 5)((-3) + 2) = (-8)(-1) = 8 > 0$ (positive)
• Test $x = 0$ (pick any value between $-2$ and $5$): $(0 - 5)(0 + 2) = (-5)(2) = -10 < 0$ (negative)
• Test $x = 6$ (pick any value greater than $5$): $(6 - 5)(6 + 2) = (1)(8) = 8 > 0$ (positive)

Step 5: Determine the solution for the inequality based on the signs:

The expression $x^2 - 3x - 10$ is non-positive ($\leq 0$) in the interval $[-2, 5]$. Therefore, the solution to the inequality is: $-2 \leq x \leq 5$

In interval notation: $[-2, 5]$ or in set notation: $\{x \mid -2 \leq x \leq 5\}$.