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

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Question 263330: 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.
10 - 5a2 = 7a + 9
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
-5a^2-7a+1=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aa%5E2%2Bba%2Bc=0 (in our case -5a%5E2%2B-7a%2B1+=+0) has the following solutons:

a%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=%28-7%29%5E2-4%2A-5%2A1=69.

Discriminant d=69 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--7%2B-sqrt%28+69+%29%29%2F2%5Ca.

a%5B1%5D+=+%28-%28-7%29%2Bsqrt%28+69+%29%29%2F2%5C-5+=+-1.53066238629181
a%5B2%5D+=+%28-%28-7%29-sqrt%28+69+%29%29%2F2%5C-5+=+0.130662386291807

Quadratic expression -5a%5E2%2B-7a%2B1 can be factored:
-5a%5E2%2B-7a%2B1+=+-5%28a--1.53066238629181%29%2A%28a-0.130662386291807%29
Again, the answer is: -1.53066238629181, 0.130662386291807. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-5%2Ax%5E2%2B-7%2Ax%2B1+%29