SOLUTION: Use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one ration

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Question 247788: Use the discriminant to determine whether the following equations have solutions that are: two different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginary solutions.
9x2 + 4x - 5 = 0
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
9x2 + 4x - 5 = 0
see below for help on discriminant
imaginary solutions would have i sqrt(-1)
irrational would have the sqrt sign
http://en.wikipedia.org/wiki/Irrational_number
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B4x%2B-5+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%284%29%5E2-4%2A9%2A-5=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+196+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+196+%29%29%2F2%5C9+=+0.555555555555556
x%5B2%5D+=+%28-%284%29-sqrt%28+196+%29%29%2F2%5C9+=+-1

Quadratic expression 9x%5E2%2B4x%2B-5 can be factored:
9x%5E2%2B4x%2B-5+=+9%28x-0.555555555555556%29%2A%28x--1%29
Again, the answer is: 0.555555555555556, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B4%2Ax%2B-5+%29