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Question

Given p-2 is one of the roots of the quadratic equation #x^2# - m#x# + 9 #-p^2# = 0 . Express p in terms of m.

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Answer 1 · 2017-08-27T12:00:10

#p = frac(2 m + 13)(m + 4)#

#p - 2# is a root of #f(x) = x^(2) - m x + 9 - p^(2)#.

So #(p - 2)^(2) - m (p - 2) + 9 - p^(2) = 0#.

Let's expand the parentheses:

#Rightarrow p^(2) - 4 p + 4 - m p + 2 m + 9 - p^(2) = 0#

Then, let's simplify the equation:

#Rightarrow p^(2) - p^(2) - 4 p - m p + 2 m + 9 + 4 = 0#

#Rightarrow - p (4 + m) + 2 m + 13 = 0#

Now, let's solve for #p#:

#Rightarrow - p (4 + m) = - (2 m + 13)#

#Rightarrow - p = - frac(2 m + 13)(4 + m)#

#therefore p = frac(2 m + 13)(m + 4)#