Question

- x^3+675x-(15+x)(225-15x+x^2)>0​

Answer 1

Ответ:

$x < 15 (\frac{ - 1 - \sqrt{3} }{2} ) \\15 (\frac{ - 1 + \sqrt{3} }{2} ) < x < 15$

Объяснение:

преобразуем

$- {x}^{3} + 3 \times {15}^{2} x - ( {15}^{3} + {x}^{3} ) > 0 \\ - 2 {x}^{3} + 2 \times {15}^{2}x + {15}^{2} x - {15}^{3} > 0 \\ - 2x( {x}^{2} - {15}^{2} ) + {15}^{2} (x - 15) > 0 \\ - 2x(x - 15)(x + 15) + {15}^{2} (x - 15) > 0 \\ (x - 15)( {15}^{2} - 2x(x + 15)) > 0 \\ (x - 15)( 2x(x + 15) - {15}^{2} ) < 0 \\ (x - 15)(2 {x}^{2} + 30x - 225) < 0 \\ 2 {x}^{2} + 30x - 225 = 0 \\ d = {30}^{2} + 4 \times 2 \times 225 = 12\times 225 \\ \sqrt{d} = 2 \times 15 \sqrt{3} = 30 \sqrt{3} \\ x_{1} = \frac{ - 30 + 30 \sqrt{3} }{4} = 15 (\frac{ - 1 + \sqrt{3} }{2} ) \\ x_{2}= 15 (\frac{ - 1 - \sqrt{3} }{2} )$

поэтому можно записать

$(x - 15)(x - 15 (\frac{ - 1 - \sqrt{3} }{2} ))(x - 15 (\frac{ - 1 + \sqrt{3} }{2} )) < 0$

так как

$15 (\frac{ - 1 - \sqrt{3} }{2} ) < 15 (\frac{ - 1 + \sqrt{3} }{2} ) < 15$

с помощью метода интервалов получаем

$x < 15 (\frac{ - 1 - \sqrt{3} }{2} ) \\15 (\frac{ - 1 + \sqrt{3} }{2} ) < x < 15$

Answer 2

To solve the inequality $-x^3 + 675x - (15 + x)(225 - 15x + x^2) > 0$, you can follow these steps:

1. Simplify the expression on the right side of the inequality.
2. Factor the expression if possible.
3. Find the critical points (where the expression equals zero).
4. Determine the intervals where the expression is positive.

Let's go through each step:

1. Simplify the expression on the right side:

$$-(15 + x)(225 - 15x + x^2) = -15(225 - 15x + x^2) - x(225 - 15x + x^2) = -3375 + 225x - 15x^2 - 225x + 15x^3 - x^3 = -x^3 + 15x^3 - 15x^2 - 3375$$

2. Now, the inequality becomes:

$$- x^3 + 675x - (x^3 - 15x^2 - 3375) > 0$$

3. Combine like terms:

$$15x^3 - 15x^2 + 675x - 3375 > 0$$

4. Factor out the common factor of 15:

$$15(x^3 - x^2 + 45x - 225) > 0$$

Now, we need to find the critical points where $x^3 - x^2 + 45x - 225 = 0$. To do this, we can use a numerical method or a graphing calculator to find the roots of the equation. Once you find the critical points, you can test intervals on the number line to see where the expression is greater than zero.

The inequality $15(x^3 - x^2 + 45x - 225) > 0$ will be satisfied in the intervals where the expression is positive. Remember to include the critical points in the intervals where the expression is positive because it's greater than zero, not greater than or equal to zero.

Please note that solving this polynomial equation exactly can be a bit challenging, so using numerical methods or a calculator might be the most practical approach to find the critical points.