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 281671: 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.
x2 + 10x + 19 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

From x%5E2%2B10x%2B19 we can see that a=1, b=10, and c=19


D=b%5E2-4ac Start with the discriminant formula.


D=%2810%29%5E2-4%281%29%2819%29 Plug in a=1, b=10, and c=19


D=100-4%281%29%2819%29 Square 10 to get 100


D=100-76 Multiply 4%281%29%2819%29 to get %284%29%2819%29=76


D=24 Subtract 76 from 100 to get 24


Since the discriminant is greater than zero, and isn't a perfect square, this means that there are two real and irrational solutions.