Question

8b ( 3b - 10) < (56 - 8) ²

Answer 1

вот ответ 100% правильно

Answer 2

To solve the inequality $8b(3b - 10) < (56 - 8)^2$, you can follow these steps:

1. Simplify both sides of the inequality:

   $8b(3b - 10) < (48)^2$

2. Expand the left side of the inequality:

   $24b^2 - 80b < 2304$

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

   $24b^2 - 80b - 2304 < 0$

4. To solve this quadratic inequality, you can first find the roots of the associated quadratic equation:

   $24b^2 - 80b - 2304 = 0$

   You can simplify this equation by dividing all terms by their greatest common divisor, which is 8:

   $3b^2 - 10b - 288 = 0$

5. Solve this quadratic equation using the quadratic formula:

   $b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

   In this case, $a = 3$, $b = -10$, and $c = -288$. Plugging these values into the formula:

   $b = \frac{10 \pm \sqrt{(-10)^2 - 4(3)(-288)}}{2(3)}$

   $b = \frac{10 \pm \sqrt{100 + 3456}}{6}$

   $b = \frac{10 \pm \sqrt{3556}}{6}$

6. Now, calculate the two possible values for $b$:

   $b_1 = \frac{10 + \sqrt{3556}}{6}$

   $b_2 = \frac{10 - \sqrt{3556}}{6}$

7. These two values of $b$ represent the points where the quadratic equation $24b^2 - 80b - 2304 = 0$ intersects the x-axis. You can use these points to determine the intervals on the number line where the quadratic inequality is satisfied.

8. Test each interval between the roots of the quadratic equation by choosing a test point within each interval and checking if the inequality $24b^2 - 80b - 2304 < 0$ holds true.

   For example, if you choose $b = 0$ as a test point, you get:

   $24(0)^2 - 80(0) - 2304 < 0$

   $-2304 < 0$

   This is true, so the interval $(-\infty, b_2)$ is part of the solution.

   You can similarly test the other interval $(b_1, \infty)$ to find that it is also part of the solution.

9. Therefore, the solution to the original inequality $8b(3b - 10) < (56 - 8)^2$ is:

   $-\infty < b < \frac{10 - \sqrt{3556}}{6}$ or $\frac{10 + \sqrt{3556}}{6} < b < \infty$