SOLUTION: Hi, i am having trouble finding the answer to this: The following equations have repeated roots. find the value(s) of the constant p leaving your answer(s) in exact form. (p

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Question 159293: Hi,
i am having trouble finding the answer to this:
The following equations have repeated roots. find the value(s) of the constant p leaving your answer(s) in exact form.
(p-1)x^2 + 6x + (p+7) = 0
i have been trying to figure it out for a while now and i have looked in the text book and the answers are -8 and 2, but i want to know how i can get these answers.
Thanks.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Note: repeated roots simply mean that the roots are equal. This only happens when the discriminant D is zero


Looking at %28p-1%29x%5E2+%2B+6x+%2B+%28p%2B7%29+=+0+, we see that a=p-1, b=6 and c=p%2B7


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


0=6%5E2-4%28p-1%29%28p%2B7%29 Plug in D=0 (since the roots repeat themselves), a=p-1, b=6 and c=p%2B7


0=36-4%28p-1%29%28p%2B7%29 Square 6 to get 36


0=36-4%28p%5E2%2B6p-7%29 FOIL


0=36-4p%5E2-24p%2B28 Distribute


0=-4p%5E2-24p%2B64 Combine like terms.


Notice we have a quadratic equation in the form of ap%5E2%2Bbp%2Bc where a=-4, b=-24, and c=64


Let's use the quadratic formula to solve for p


p+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


p+=+%28-%28-24%29+%2B-+sqrt%28+%28-24%29%5E2-4%28-4%29%2864%29+%29%29%2F%282%28-4%29%29 Plug in a=-4, b=-24, and c=64


p+=+%2824+%2B-+sqrt%28+%28-24%29%5E2-4%28-4%29%2864%29+%29%29%2F%282%28-4%29%29 Negate -24 to get 24.


p+=+%2824+%2B-+sqrt%28+576-4%28-4%29%2864%29+%29%29%2F%282%28-4%29%29 Square -24 to get 576.


p+=+%2824+%2B-+sqrt%28+576--1024+%29%29%2F%282%28-4%29%29 Multiply 4%28-4%29%2864%29 to get -1024


p+=+%2824+%2B-+sqrt%28+576%2B1024+%29%29%2F%282%28-4%29%29 Rewrite sqrt%28576--1024%29 as sqrt%28576%2B1024%29


p+=+%2824+%2B-+sqrt%28+1600+%29%29%2F%282%28-4%29%29 Add 576 to 1024 to get 1600


p+=+%2824+%2B-+sqrt%28+1600+%29%29%2F%28-8%29 Multiply 2 and -4 to get -8.


p+=+%2824+%2B-+40%29%2F%28-8%29 Take the square root of 1600 to get 40.


p+=+%2824+%2B+40%29%2F%28-8%29 or p+=+%2824+-+40%29%2F%28-8%29 Break up the expression.


p+=+%2864%29%2F%28-8%29 or p+=++%28-16%29%2F%28-8%29 Combine like terms.


p+=+-8 or p+=+2 Simplify.


So the answers are p+=+-8 or p+=+2


This means that if p+=+-8 or p+=+2, then the equation %28p-1%29x%5E2+%2B+6x+%2B+%28p%2B7%29+=+0+ will have repeated roots.