Question

Решите уравнение (x^2-3x-4) ^2 +2(x-5)(x+2)+12=0

Answer 1

Решение:
$( x^{2} - 3x - 4) ^{2} + 2*(x - 5)* (x + 2) + 12 = 0$
$( x^{2} - 3x - 4) ^{2} + 2*( x^{2} - 3x - 10) + 12 = 0$
$( x^{2} - 3x - 4) ^{2} + 2*( (x^{2} - 3x - 4) - 6) + 12 = 0$
$( x^{2} - 3x - 4) ^{2} + 2* (x^{2} - 3x - 4) - 2*6 + 12 = 0$
$( x^{2} - 3x - 4) ^{2} + 2* (x^{2} - 3x - 4) = 0$
Пусть  $x^{2} - 3x - 4 = t$, тогда 
$t^{2} + 2t = 0$
$t = 0$ или $t = - 2$
Получили, что 
1)  $x^{2} - 3x - 4 = 0$ или 2)  $x^{2} - 3x - 4 = -2$
Решим отдельно каждое получившееся уравнение:
(1): $x^{2} - 3x - 4 = 0, D = 25, x_{1} = 4, x_{2} = - 1$
(2):  $x^{2} - 3x - 4 = - 2, x^{2} - 3x - 2 = 0, D = 17, x_{1} = \frac{3 + \sqrt{17} }{2} , x_{2} = \frac{3 - \sqrt{17} }{2}$
Ответ:
-1; 4;  $\frac{3 - \sqrt{17} }{2}, \frac{3 + \sqrt{17} }{2}$.

Answer 2

To solve the equation $(x^2-3x-4)^2 + 2(x-5)(x+2) + 12 = 0$, we'll follow these steps:

1. Expand the expression $(x^2-3x-4)^2$ to simplify the equation.
2. Simplify the expression $2(x-5)(x+2)$.
3. Combine like terms.
4. Solve the resulting quadratic equation.

Let's start step by step:

1. Expand the expression $(x^2-3x-4)^2$: $(x^2-3x-4)^2 = (x^2-3x-4)(x^2-3x-4)$ $= x^2(x^2) - x^2(3x) - x^2(4) - 3x(x^2) + 3x(3x) + 3x(4) - 4(x^2) + 4(3x) + 4(4)$ $= x^4 - 3x^3 - 4x^2 - 3x^3 + 9x^2 + 12x - 4x^2 + 12x + 16$ $= x^4 - 6x^3 + 5x^2 + 24x + 16$

2. Simplify the expression $2(x-5)(x+2)$: $2(x-5)(x+2) = 2(x^2 - 3x - 10)$ $= 2x^2 - 6x - 20$

3. Now, the equation becomes: $x^4 - 6x^3 + 5x^2 + 24x + 16 + 2x^2 - 6x - 20 + 12 = 0$

4. Combine like terms: $x^4 - 6x^3 + 5x^2 + 2x^2 + 24x - 6x + 16 - 20 + 12 = 0$ $x^4 - 6x^3 + 7x^2 + 18x + 8 = 0$

Unfortunately, this is a quartic equation, which means it's not directly solvable using simple algebraic methods. In general, quartic equations are solved using numerical methods or factoring if possible. Since factoring doesn't seem straightforward here, we'll have to resort to numerical methods or graphing to approximate the solutions.

If you need to find the approximate solutions, you can use numerical methods such as the Newton-Raphson method, or you can use software like calculators or computer algebra systems to find the solutions.