Question

How do you find the discriminant and how many and what type of solutions does `4x^2-7x+2=0` have?

Answer 1

We have two conjugate irrational numbers of type `a+-sqrtb`.

Explanation:

The discriminant of a quadratiic equation `ax^2+bx+c=0` is `b^2-4ac`. Assume that coefficients `a`, `b` and `c` are all integers, then

1. If `b^2-4ac` is equal to zero, we have just one solution
2. If `b^2-4ac>0` and it is square of a rational number, we have two solutions, each a rational number.
3. If `b^2-4ac>0` and it is not a square of a rational number, we have two solutions, each root being a real number but not rational number. Tey will be of type `a+-sqrtb`, two conjugate irrational numbers.
4. If `b^2-4ac<0`, we have two solutions, which are two complex conjugate numbers.

Here in `4x^2-7x+2-0`, we have `a=4,b=-7` and `c=2` and hence dicriminant is `(-7)^2-4*4*2=49-32=17`

we have two conjugate irrational numbers of type `a+-sqrtb`