for an equation such as y = x2 +2x -3, you can solve for roots with the quadratic formula. Or you can factor to get the roots like so y = (x+3)(x-1) therefore the roots are -3 & 1.
to complete the square, you write the alternative equation as (x+h)2 + k . Considering the same equation as above in the form ax2 + bx + c where a=1, b=2, & c=-3, h is equal to 1/2 of b (2)., then you set the equations equal to one another to find k.
x2 + 2x - 3 = (x+1)2 + k expanding we get x2 + 2x - 3 = x2 + 2x + 1 + k
Solving for k, we get k = -3-4 = -4. So, the completed square is (x+1)2 - 4
In high school, factoring cube roots is usually done with a polynomial that can be factored. For example,
Let's consider y = x3 + 2x -2x -4. It can be grouped into y = x2(x+2) -2(x-2), now you can take (x+2) such that
y = (x+2)(x2-2) = (x+2)(x-sqrt(2))(x+sqrt(2)) then the roots are obvious!
If you know one of the roots (r) of a cubic equation by inspection, you can divide the polynominal by( x-r) to reduce it to a quadratic that can be solved easily!
