Question

4x2 - 4x - 34 ≥ 9x2 - 3x - 56

Answer 1

Ответ:

4x^2 - 9x^2 - 4x + 3x - 34 + 56 >= 0

- 5x^2 - x + 22 >= 0

5x^2 + x - 22 <= 0

ввдем функцию y = 5x^2 + x - 22, функция является парабалой, ветви направлены вверх

нули функции: y=0

5x^2 + x - 22 = 0

D = 1 + 440 = 441

x1 = -1 - 21 / 10 = 2,2

x2 = -1 + 21 / 10 = 2

y=0,при x принадлежащем [2; 2,2]

Answer 2

To solve the inequality $4x^2 - 4x - 34 \geq 9x^2 - 3x - 56$, follow these steps:

1. Subtract the right-hand side from the left-hand side:

   $4x^2 - 4x - 34 - (9x^2 - 3x - 56) \geq 0$

   Simplify each term:

   $-5x^2 - x + 22 \geq 0$

2. Factor the quadratic if possible:

   The quadratic $5x^2 + x - 22$ doesn't factor easily, so we'll use the quadratic formula:

   The quadratic formula states that for a quadratic equation $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

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

   In this case, the coefficients are $a = -5$, $b = -1$, and $c = 22$.

   $x = \frac{1 \pm \sqrt{(-1)^2 - 4(-5)(22)}}{2(-5)}$

   $x = \frac{1 \pm \sqrt{1 + 440}}{-10}$

   $x = \frac{1 \pm \sqrt{441}}{-10}$

   $x = \frac{1 \pm 21}{-10}$

   So, the solutions for the quadratic are $x = -2$ and $x = \frac{11}{5}$.

3. Determine the sign of the quadratic in the intervals determined by the solutions:

   We have two critical points, $x = -2$ and $x = \frac{11}{5}$. Test a value in each interval to determine the sign of the quadratic:

   • Choose a value less than -2, e.g., -3: $f(-3) = (-5)(-3)^2 - (-3) + 22 = 55 + 3 + 22 = 80$ Since $80$ is positive, the quadratic is positive in the interval $(-\infty, -2)$.

   • Choose a value between -2 and $\frac{11}{5}$, e.g., 0: $f(0) = (-5)(0)^2 - (0) + 22 = 22$ Since $22$ is positive, the quadratic is positive in the interval $(-2, \frac{11}{5})$.

   • Choose a value greater than $\frac{11}{5}$, e.g., $2$: $f\left(\frac{11}{5}\right) = (-5)\left(\frac{11}{5}\right)^2 - \frac{11}{5} + 22 = -\frac{121}{5} - \frac{11}{5} + 22 = \frac{22}{5} - \frac{132}{5} = -\frac{110}{5} = -22$ Since $-22$ is negative, the quadratic is negative in the interval $\left(\frac{11}{5}, \infty\right)$.

4. Determine the valid solution set:

   The inequality is $5x^2 + x - 22 \geq 0$