I've been trying to factorize this quadratic equation to find it's roots. (I know how to use discriminant but this has to be solved using factorization)
I asssumed the middle term will split in terms of √5 and then I'm stuck at another quadratic. Can't figure out it's roots without using discriminant.
Any help?
Best Answer
You could use the quadratic complement, that is, divide the equation by the coefficient of $x^2$, then add and subtract the square of half of the new coefficient of $x$.
So let us firstly divide the given equation by $2$, yielding $$x^2-\frac{\sqrt 5}{2}x-1=0.$$ Then we add and subtract $5/16$ to the left member: $$x^2-\frac{\sqrt 5}{2}x+\frac{5}{16}-\frac{5}{16}-1=0.$$ Tidying up gives $$\Big(x-\frac{\sqrt 5}{4}\Big)^2=\frac{21}{16}.$$ Can you finish from here?