Question

(x+3)(x+6)(x+9)(x+12)=45,5625

Answer 1

Три корня. Решение во вложении.

Answer 2

To solve the equation $(x+3)(x+6)(x+9)(x+12) = 45.5625$, we can start by simplifying the left-hand side of the equation first. Notice that the factors on the left-hand side are consecutive terms with a common difference of 3, which suggests that we can use a substitution to make the equation easier to handle.

Let's make a substitution to simplify the equation:

Let $y = x + 6$. This substitution will simplify the equation, making it a quadratic equation.

Now, the equation becomes:

$(y - 3)(y)(y + 3)(y + 6) = 45.5625$

Now, let's simplify this:

$(y^2 - 3y)(y^2 + 6y) = 45.5625$

Expand the equation:

$y^4 + 6y^3 - 3y^3 - 18y^2 = 45.5625$

Combine like terms:

$y^4 + 3y^3 - 18y^2 - 45.5625 = 0$

The equation has been simplified to a quartic equation in terms of $y$. Now, we can attempt to solve it. However, solving quartic equations can be complex and might require numerical methods or other advanced techniques. In this case, it's not immediately apparent how to find the exact solutions.

If you are looking for approximate solutions, you can use numerical methods like numerical root-finding algorithms or graphing the equation to find approximate values of $y$ (remember, $y = x + 6$), which will give you the approximate solutions for $x$. Alternatively, you can use a computer algebra system (CAS) or calculator capable of solving quartic equations to find the exact solutions.

Keep in mind that the result of $45.5625$ suggests that the solutions might involve complex numbers, as the quartic equation can have complex roots.