How do you find the roots, real and imaginary, of #y= 2 x(x - 4) -(2x-1)^2 # using the quadratic formula?
1 Answer
Feb 2, 2016
We have two real roots:
Explanation:
First of all, let's expand and simplify the expression.
#y = 2x(x-4) - (2x-1)^2 #
# color(white)(x) = 2x^2 - 8x - (4x^2 - 4x + 1)#
... use the formula
# color(white)(x) = 2x^2 - 8x - 4x^2 + 4x - 1#
# color(white)(x) = - 2x^2 - 4x - 1#
Now, we are ready to use the quadratic formula.
The formula is
#x = (-b +- sqrt(b^2 - 4ac))/(2a)#
In our case,
#x = (4 +- sqrt((-4)^2 - 4 * (-2) * (-1)))/(2 * (-2)) = (4 +- sqrt(16 - 8))/(-4) = (4 +- sqrt(8))/(-4) = (4 +- 2 sqrt(2))/(-4) = -1 +- sqrt(2)/2#
So, we have two real solutions:
#x_1 = -1 + sqrt(2)/2 ~~ - 0.29#
and
#x_2 = -1 - sqrt(2)/2 ~~ -1.71#