We begin with y=2x^2-x+(2x-1)^2. Our first step is to simplify our equation until it cannot be reduced any further. Then we move on to the quadratic formula.
y=2x^2-x+(2x-1)^2 can be changed to y=2x^2-x+((2x-1)*(2x+1)). That becomes y=2x^2-x+4x^2-2x-2x+1 or y=2x^2-x+4x^2-4x+1. Now we combine like terms like 2x^2+4x^2 and -x-5x. That gives us y=6x^2-5x+1.
Now we can move on to the quadratic equation, which is (-b+-sqrt(b^2-4*a*c))/(2a) where those variables come from ax^2+bx+c. In our case a=6, b=-5, and c=1. Now we just plug in our value to the formulla: (5+-sqrt(-5^2-4*6*1))/(2*6). This can be simplified to (5+-sqrt(25-24))/(12) or (5+-1)/12. That becomes 4/12 or 6/12, which can be sipmlified to 1/3 and 1/2, respectivley. Those are our roots, and we can graph the original equation and look at the x-intercepts (roots).
graph{y=6x^2-5x+1}
And they are! Nice work!
