How do you find the discriminant and how many and what type of solutions does #x^2+3x+7=0# have?
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
- Divide by
#alpha#both sides, getting
- Extract a square root from both side using positive and negative values of a square root, getting
#beta#from both sides, getting
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:
In a particular case of this problem with an equation
the coefficients are:
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.