# How do you find the discriminant and how many and what type of solutions does #x^2+3x+7=0# have?

##### 1 Answer

Let's get some theoretical background first.

Assume you have to solve a general *quadratic* equation

You have no recollection of the formula for its solution (like most people). How should you solve it?

Let's apply the logic.

If your equation looks like this

the solution is easy:

- You subtract
#gamma# from both sides, getting

#alpha(x+beta)^2=-gamma# - Divide by
#alpha# both sides, getting

#(x+beta)^2=-gamma/alpha - Extract a square root from both side using positive and negative values of a square root, getting

#x+beta=sqrt(-gamma/alpha)# - Subtract
#beta# from both sides, getting

#x_(1,2)=+-sqrt(-gamma/alpha)-beta#

Our general equation is not of the form we can easily solve, as above, but can be transformed into this form.

Let's equate two forms and find the necessary

Open the parenthesis on the right:

Now you can determine

from which follows:

Now we use the formula that expressed the solution in terms of

The most involved part of this expression is under the square root, it's called *discriminant* of this quadratic equation:

If

If

Finally, if

In a particular case of this problem with an equation

the coefficients are:

The discriminant

It's negative. Therefore, there are no real solutions.

A more detailed explanation of topics touched in this answer can be found on Unizor by following the menu items *Algebra - Quadratic Equations*.