Question

1) (x + 8)2 – x2 <11x;​

Answer 1

Ответ:

16+2х-х2-11х<0

здесь получаеться -х2-9х+16<0

если это разделить на - получаеться

х2+9х-16>0

нет такого числа чтоб при сложении получилось 9 а при умножении 16 поэтому ответ пустое множество

Answer 2

To solve the inequality (x + 8)^2 - x^2 < 11x, you can follow these steps:

1. Expand the squares:

   (x^2 + 16x + 64) - x^2 < 11x

2. Simplify the equation:

   16x + 64 - x^2 < 11x

3. Move all terms to one side of the inequality:

   16x + 64 - x^2 - 11x < 0

4. Combine like terms:

   5x - x^2 + 64 < 0

5. Rearrange the terms:

   -x^2 + 5x + 64 < 0

6. To solve this quadratic inequality, you can use a sign chart or factoring. Factoring is a bit tricky in this case, so we'll use a sign chart.

   First, find the critical points by setting the expression equal to zero:

   -x^2 + 5x + 64 = 0

   To solve this quadratic equation, you can use the quadratic formula:

   x = (-b ± √(b^2 - 4ac)) / (2a)

   In this case, a = -1, b = 5, and c = 64:

   x = (-5 ± √(5^2 - 4(-1)(64))) / (2(-1)) x = (-5 ± √(25 + 256)) / (-2) x = (-5 ± √281) / (-2)

   The two solutions are:

   x₁ = (-5 + √281) / (-2) x₂ = (-5 - √281) / (-2)

7. Now, create a sign chart using these critical points:

   Place the critical points on a number line:

   |------------------x₁--------------------x₂------------------|

8. Test the regions created by the critical points with any value from each region to determine the sign of the expression:

   • Pick a value less than x₁, for example, x = -10: -(-10)^2 + 5(-10) + 64 = -100 - 50 + 64 = -86 This region is negative.

   • Pick a value between x₁ and x₂, for example, x = 0: -(0)^2 + 5(0) + 64 = 64 This region is positive.

   • Pick a value greater than x₂, for example, x = -20: -(-20)^2 + 5(-20) + 64 = -400 - 100 + 64 = -436 This region is negative.

9. Now, determine the solution based on the sign chart:

   The expression is negative in the regions to the left and right of the critical points. So, the solution to the inequality is:

   x < (-5 - √281) / (-2) or x > (-5 + √281) / (-2)

You can simplify this further if needed.