X²+4x-21>0

Question 1

Задача Задача №86397

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Аноним

05 декабря 2023 12:14 · публичный вопрос

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Аноним 05 декабря 2023 12:14

X²+4x-21>0

Answer

To solve the inequality x² + 4x - 21 > 0, you can use different methods such as factoring, the quadratic formula, or the method of graphing.

Factoring Method: Let's first factor the quadratic equation x² + 4x - 21 > 0:

(x + 7)(x - 3) > 0

Now we have two factors, (x + 7) and (x - 3), and we want their product to be greater than zero.

(a) If both factors are positive (+)(+), the product will be positive. x + 7 > 0 and x - 3 > 0 x > -7 and x > 3 Since x > 3 is more restrictive, we take that condition only.

(b) If both factors are negative (-)(-), the product will also be positive. x + 7 < 0 and x - 3 < 0 x < -7 and x < 3 Since x < -7 is more restrictive, we take that condition only.

Therefore, the solution to x² + 4x - 21 > 0 is x < -7.

Quadratic Formula Method: Another way to solve the inequality is by applying the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For x² + 4x - 21 > 0, a = 1, b = 4, and c = -21.

Applying the quadratic formula, we have: x = (-4 ± √(4² - 4(1)(-21))) / (2(1)) x = (-4 ± √(16 + 84)) / 2 x = (-4 ± √100) / 2 x = (-4 ± 10) / 2

This yields two solutions: x₁ = (-4 + 10) / 2 = 3 x₂ = (-4 - 10) / 2 = -7

Now, we can test each solution to determine the intervals in which the inequality holds true:

For x < -7: x² + 4x - 21 = (-7)² + 4(-7) - 21 = 49 - 28 - 21 = 0, which is not greater than 0. For x > 3: x² + 4x - 21 = (3)² + 4(3) - 21 = 9 + 12 - 21 = 0, which is not greater than 0.

Therefore, the solution is x < -7.

Graphical Method: To graph the inequality x² + 4x - 21 > 0, you can start by finding the x-intercepts by setting x² + 4x - 21 = 0. Solving this equation, you will get x = -7 and x = 3. These are the critical points.

Next, you can plot these critical points on a number line and test a value in each interval.

For x < -7 (e.g., x = -8): x² + 4x - 21 = (-8)² + 4(-8) - 21 = 64 - 32 - 21 = 11, which is greater than 0.

For -7 < x < 3 (e.g., x = 0): x² + 4x - 21 = (0)² + 4(0) - 21 = -21, which is not greater than 0.

For x > 3 (e.g., x = 4): x² + 4x - 21 = (4)² + 4(4) - 21 = 16 + 16 - 21 = 11, which is greater than 0.

Therefore, the solution is x < -7.

Question 2

x²+4x-21>0

Answer

To solve the inequality x² + 4x - 21 > 0, you can use different methods such as factoring, the quadratic formula, or the method of graphing. 1. Factoring Method: Let's first factor the quadratic equation x² + 4x - 21 > 0: (x + 7)(x - 3) > 0 Now we have two factors, (x + 7) and (x - 3), and we want their product to be greater than zero. (a) If both factors are positive (+)(+), the product will be positive. x + 7 > 0 and x - 3 > 0 x > -7 and x > 3 Since x > 3 is more restrictive, we take that condition only. (b) If both factors are negative (-)(-), the product will also be positive. x + 7 < 0 and x - 3 < 0 x < -7 and x < 3 Since x < -7 is more restrictive, we take that condition only. Therefore, the solution to x² + 4x - 21 > 0 is x < -7. 2. Quadratic Formula Method: Another way to solve the inequality is by applying the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) For x² + 4x - 21 > 0, a = 1, b = 4, and c = -21. Applying the quadratic formula, we have: x = (-4 ± √(4² - 4(1)(-21))) / (2(1)) x = (-4 ± √(16 + 84)) / 2 x = (-4 ± √100) / 2 x = (-4 ± 10) / 2 This yields two solutions: x₁ = (-4 + 10) / 2 = 3 x₂ = (-4 - 10) / 2 = -7 Now, we can test each solution to determine the intervals in which the inequality holds true: For x < -7: x² + 4x - 21 = (-7)² + 4(-7) - 21 = 49 - 28 - 21 = 0, which is not greater than 0. For x > 3: x² + 4x - 21 = (3)² + 4(3) - 21 = 9 + 12 - 21 = 0, which is not greater than 0. Therefore, the solution is x < -7. 3. Graphical Method: To graph the inequality x² + 4x - 21 > 0, you can start by finding the x-intercepts by setting x² + 4x - 21 = 0. Solving this equation, you will get x = -7 and x = 3. These are the critical points. Next, you can plot these critical points on a number line and test a value in each interval. For x < -7 (e.g., x = -8): x² + 4x - 21 = (-8)² + 4(-8) - 21 = 64 - 32 - 21 = 11, which is greater than 0. For -7 < x < 3 (e.g., x = 0): x² + 4x - 21 = (0)² + 4(0) - 21 = -21, which is not greater than 0. For x > 3 (e.g., x = 4): x² + 4x - 21 = (4)² + 4(4) - 21 = 16 + 16 - 21 = 11, which is greater than 0. Therefore, the solution is x < -7.