0,5x^2 +8x + 24 <0

Question 1

Задача Задача №72526

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Аноним

29 ноября 2023 21:18 · публичный вопрос

А

Аноним 29 ноября 2023 21:18

0,5x^2 +8x + 24 <0

Answer

To solve the inequality 0.5x^2 + 8x + 24 < 0, we can follow these steps:

Step 1: Write the quadratic equation in standard form, setting it equal to 0: 0.5x^2 + 8x + 24 = 0

Step 2: Determine the discriminant to see if the quadratic has real solutions. The discriminant, denoted as Δ, is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

In this case, a = 0.5, b = 8, and c = 24. The discriminant is: Δ = (8)^2 - 4(0.5)(24) Δ = 64 - 48 Δ = 16

Since the discriminant is positive (Δ > 0), the quadratic has two real solutions.

Step 3: Find the x-intercepts (the values of x where the quadratic equation crosses the x-axis). This can be done by factoring, completing the square, or using the quadratic formula. In this case, we'll use the quadratic formula since factoring is not possible for this equation.

The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values of a, b, and c into the formula: x = (-8 ± √(8^2 - 4(0.5)(24))) / (2(0.5)) x = (-8 ± √(64 - 48)) / 1 x = (-8 ± √16)

Simplifying further: x = (-8 ± 4)

This gives us two solutions: x1 = -8 + 4 = -4 x2 = -8 - 4 = -12

Therefore, the x-intercepts are x = -4 and x = -12.

Step 4: Determine the sign of the quadratic equation in each interval using a test point from each interval.

Choose a test point in the interval (-∞, -12) such as x = -13: 0.5(-13)^2 + 8(-13) + 24 < 0 (0.5)(169) - 104 + 24 < 0 84.5 - 104 + 24 < 0 4.5 - 104 + 24 < 0 -75.5 < 0

Since -75.5 is less than 0, the inequality is true for the interval (-∞, -12).

Choose a test point in the interval (-12, -4) such as x = -8: 0.5(-8)^2 + 8(-8) + 24 < 0 (0.5)(64) - 64 + 24 < 0 32 - 64 + 24 < 0 -8 < 0

Since -8 is less than 0, the inequality is true for the interval (-12, -4).

Choose a test point in the interval (-4, ∞) such as x = 0: 0.5(0)^2 + 8(0) + 24 < 0 0 + 0 + 24 < 0 24 < 0

Since 24 is not less than 0, the inequality is false for the interval (-4, ∞).

Step 5: Write the final solution as an interval notation:

The solution to the inequality is (-∞, -12) ∪ (-4, ∞).

Question 2

0,5x^2 +8x + 24 <0

Answer

To solve the inequality 0.5x^2 + 8x + 24 < 0, we can follow these steps: Step 1: Write the quadratic equation in standard form, setting it equal to 0: 0.5x^2 + 8x + 24 = 0 Step 2: Determine the discriminant to see if the quadratic has real solutions. The discriminant, denoted as Δ, is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, a = 0.5, b = 8, and c = 24. The discriminant is: Δ = (8)^2 - 4(0.5)(24) Δ = 64 - 48 Δ = 16 Since the discriminant is positive (Δ > 0), the quadratic has two real solutions. Step 3: Find the x-intercepts (the values of x where the quadratic equation crosses the x-axis). This can be done by factoring, completing the square, or using the quadratic formula. In this case, we'll use the quadratic formula since factoring is not possible for this equation. The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a) Substituting the values of a, b, and c into the formula: x = (-8 ± √(8^2 - 4(0.5)(24))) / (2(0.5)) x = (-8 ± √(64 - 48)) / 1 x = (-8 ± √16) Simplifying further: x = (-8 ± 4) This gives us two solutions: x1 = -8 + 4 = -4 x2 = -8 - 4 = -12 Therefore, the x-intercepts are x = -4 and x = -12. Step 4: Determine the sign of the quadratic equation in each interval using a test point from each interval. Choose a test point in the interval (-∞, -12) such as x = -13: 0.5(-13)^2 + 8(-13) + 24 < 0 (0.5)(169) - 104 + 24 < 0 84.5 - 104 + 24 < 0 4.5 - 104 + 24 < 0 -75.5 < 0 Since -75.5 is less than 0, the inequality is true for the interval (-∞, -12). Choose a test point in the interval (-12, -4) such as x = -8: 0.5(-8)^2 + 8(-8) + 24 < 0 (0.5)(64) - 64 + 24 < 0 32 - 64 + 24 < 0 -8 < 0 Since -8 is less than 0, the inequality is true for the interval (-12, -4). Choose a test point in the interval (-4, ∞) such as x = 0: 0.5(0)^2 + 8(0) + 24 < 0 0 + 0 + 24 < 0 24 < 0 Since 24 is not less than 0, the inequality is false for the interval (-4, ∞). Step 5: Write the final solution as an interval notation: The solution to the inequality is (-∞, -12) ∪ (-4, ∞).