Question

(2x - 1) (4x^2 + 2x + 1) - 8x = 3x + 4 Решите уравнение

Answer 1

(2x - 1) (4x^2 + 2x + 1) - 8x = 3x + 4
8x^3 -1 - 8x = 3x + 4
8x^3 -5 - 11x = 0
т.к. корень 5, -5 и 1, -1 не подходят, то значит действительных корней нет

Answer 2

To solve the given equation, we need to find the values of x that satisfy it. Let's solve the equation step by step:

(2x - 1)(4x^2 + 2x + 1) - 8x = 3x + 4

First, let's expand the left side of the equation:

(2x)(4x^2) + (2x)(2x) + (2x)(1) - (1)(4x^2) - (1)(2x) - (1) - 8x = 3x + 4

8x^3 + 4x^2 + 2x - 4x^2 - 2x - 1 - 8x = 3x + 4

Next, combine like terms:

8x^3 - 8x = 3x + 4

Now, let's bring all the terms to one side of the equation:

8x^3 - 8x - 3x - 4 = 0

Combine like terms again:

8x^3 - 11x - 4 = 0

At this point, we have a cubic equation. Unfortunately, cubic equations do not have a general algebraic formula like quadratic equations. So, in general, we cannot find exact solutions for this equation.

However, we can use numerical methods or graphing to find approximate solutions. One common method is using numerical approximation techniques such as the Newton-Raphson method.

Without using numerical methods, we can still observe that there is at least one real solution because the given equation is cubic, and cubic equations always have at least one real root.

To further approximate the solutions, we can use graphing software or calculators. Plotting the graph of y = 8x^3 - 11x - 4 and finding the x-intercept will give us an approximate solution.

Keep in mind that the actual values of x satisfying the equation might not be easily expressible in simple radicals or rational numbers, so approximate solutions are often sufficient.