Question

-x²+12x+45 < 0 решите неравенство)

Answer 1

Ответ в файле

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Answer 2

To solve the inequality -x² + 12x + 45 < 0, follow these steps:

Step 1: Factor the quadratic expression. Step 2: Find the critical points by setting the expression equal to zero and solving for x. Step 3: Create a sign chart to determine the intervals where the expression is positive or negative. Step 4: Identify the solution to the inequality.

Step 1: Factor the quadratic expression To factor -x² + 12x + 45, we need to find two numbers whose sum is 12 and whose product is 45. The numbers are 15 and -3.

-x² + 12x + 45 = -(x² - 12x - 45) = -(x - 15)(x + (-3)) = -(x - 15)(x - 3)

Step 2: Find the critical points Set each factor equal to zero and solve for x:

x - 15 = 0 --> x = 15 x - 3 = 0 --> x = 3

Step 3: Create a sign chart

On the sign chart, we check the sign of the expression -x² + 12x + 45 for three intervals: (-∞, 3), (3, 15), and (15, ∞).

Intervals: (-∞, 3) | (3, 15) | (15, ∞) Test Points: 0 | 10 | 20 Sign: - | + | -

Step 4: Identify the solution to the inequality The inequality -x² + 12x + 45 < 0 is satisfied when the expression is negative. From the sign chart, we can see that the expression is negative in the interval (3, 15).

Thus, the solution to the inequality is 3 < x < 15. In interval notation, the solution is (3, 15).