To solve the given equation, we can start by simplifying both sides.
3x - 5 + 7x = 3x^2 + 7 + 11x
Combine like terms on the left side:
3x + 7x - 5 = 3x^2 + 11x + 7
10x - 5 = 3x^2 + 11x + 7
Next, move all terms to one side of the equation to set it equal to zero:
3x^2 + 11x + 7 - 10x + 5 = 0
3x^2 + x + 12 = 0
Now, the equation is in the standard form of a quadratic equation (ax^2 + bx + c = 0). To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2-4ac)) / 2a
In this case, a = 3, b = 1, and c = 12. Substituting these values into the formula:
x = (-1 ± √(1^2 - 4312)) / (2*3)
x = (-1 ± √(1 - 144)) / 6
x = (-1 ± √(-143)) / 6
Since the square root of -143 is an imaginary number, the solutions will be complex numbers:
x = (-1 ± √143i) / 6
Therefore, the solutions to the given equation are:
x = (-1 + √143i) / 6 and x = (-1 - √143i) / 6
