Giải phương trình chứa căn

To solve this equation, we need to isolate the square root terms on one side and the non-square root terms on the other side.

First, let's move the non-square root terms to the right side:

√(x+3) + √(3x+1) - √(3x^2+10x+3) = 2x - 2

Next, let's square both sides of the equation to eliminate the square roots:

(√(x+3) + √(3x+1) - √(3x^2+10x+3))^2 = (2x - 2)^2

Expanding the left side of the equation:

(x+3) + 2√((x+3)(3x+1)) - 2√((x+3)(3x^2+10x+3)) + (3x+1) - 2√((3x+1)(3x^2+10x+3)) = 4x^2 - 8x + 4

Combining like terms:

4x^2 + 6x + 4 - 2√((x+3)(3x+1)(3x^2+10x+3)) = 4x^2 - 8x + 4

Simplifying:

6x + 4 - 2√((x+3)(3x+1)(3x^2+10x+3)) = -8x

Moving the terms involving the square root to the left side:

2√((x+3)(3x+1)(3x^2+10x+3)) = -14x - 4

Squaring both sides again to eliminate the square root:

4((x+3)(3x+1)(3x^2+10x+3)) = (-14x - 4)^2

Expanding the right side:

4((x+3)(3x+1)(3x^2+10x+3)) = 196x^2 + 112x + 16

Expanding the left side:

4(9x^4 + 37x^3 + 46x^2 + 13x + 9) = 196x^2 + 112x + 16

Distributing the 4:

36x^4 + 148x^3 + 184x^2 + 52x + 36 = 196x^2 + 112x + 16

Combining like terms:

36x^4 + 148x^3 - 12x^2 - 60x + 20 = 0

Dividing through by 4:

9x^4 + 37x^3 - 3x^2 - 15x + 5 = 0

Unfortunately, this equation cannot be easily factored or solved algebraically. We would need to use numerical methods, such as graphing or using a calculator, to find the approximate solutions.