I'm a student who started self learning quadratic equations for a youth university program. I'm busy at trying to solve such equation:
$$ (1 - 4x)^2 + 9x + 7 = 2(x+3)(1-x) + (x+4)^2 $$
this is my current progress:
\begin{align} (1 - 4x)^2 + 9x + 7 &= 2(x+3)(1-x)+ (x+4)^2\\ (1 - 4x)(1 - 4x) + 9x + 7 &= (2x + 6)(1 - x) + (x + 4)(x + 4)\\ 1 - 4x - 4x + 16x^2 + 9x + 7 &= 2x - 2x^2 + 6 - 6x + x^2 + 4x + 4x + 16\\ 8 + 16x^2 + x &= 2x - x^2 + 6 - 6x + 8x + 16\\ 8 + 16x^2 + x &= 4x - x^2 + 22\\ 16x^2 + x &= 4x - x^2 + 14\\ 16x^2 &= 3x - x^2 + 14\\ 17x^2 &= 3x + 14 \end{align} The solutions to this equation are $x = 1,~x=-14/17$. So, where is my mistake? $x$ is negative, so I must be incorrect.