We begin with y=-5x^2+7x+2(2x-1)^2. In order to use the quadratic formula, we need to get this equation into standard form, which means that the format needs to look like this: ax^2+bx+c.
So, in order to change the form of our current equation, we first need to expand everything that we can. Let's start with (2x-1)^2. This becomes (2x-1)(2x-1), which is 4x^2-4x+1. Now we have -5x^2+7x+2(4x^2-4x+1)/ By multiplying out the 2, we arrive at -5x^2+7x+8x^2-8x+2.
From here, we just need to combine like terms. -5x^2+8x^2 gives us 3x^2, while 7x-8x equals -x. The 2 is left alone.
So now we have the following: y=3x^2-x+2, which is in standard form, meaning that we are good to go for using the quadratic formula!
(-b+-sqrt(b^2-4*a*c))/(2*a)
color(blue)(a=3)
color(green)(b=-1)
color(red)(c=2)
We just need to substitute the as and bs and cs for their numerical value.
The formula now becomes
(-(color(green)(-1))+-sqrt((color(green)(-1))^2-4*(color(blue)(3))*(color(red)(2))))/(2(color(blue)(3)))
Simplifying that we have (1+-sqrt(1-24))/6 or (1+-isqrt(23))/6.
That is the imaginary root for y=-5x^2+7x+2(2x-1)^2. Nice job.
