SOLUTION: if one root of the equation (p^2+2)x^2+7x+3p=0 is the reciprocal of the another then find the value of p

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: if one root of the equation (p^2+2)x^2+7x+3p=0 is the reciprocal of the another then find the value of p       Log On


   

Question 955784: if one root of the equation (p^2+2)x^2+7x+3p=0 is the reciprocal of the another then find the value of p
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Let r be one root. Then the other root will be 1%2Fr. Consequently, their product will be %28r%29%2A%281%2Fr%29+=+1. This is true if r is nonzero.
Now, let f%28x%29+=+%28p%5E2%2B2%29x%5E2%2B7x%2B3p=+ax%5E2+%2B+bx+%2B+c. => a=%28p%5E2%2B2%29 ,b=7,+c=3p
If f%28x%29+=+0, then we are given that the roots are r and 1%2Fr.

since product of roots is +c%2Fa=1, we have
=>c%2Fa=3p%2F%28p%5E2%2B2%29=1
3p=1%2A%28p%5E2%2B2%29
0=p%5E2-3p%2B2
0=p%5E2-2p-p%2B2
0=%28p%5E2-2p%29-%28p-2%29
0=p%28p-2%29-%28p-2%29
0=%28p-1%29%28p-2%29

=> highlight%28p=1%29+ or
=> highlight%28p=2%29+


since SUM OF THE ROOTS is
+r+%2B+1%2Fr+=+-b%2Fa
+r+%2B+1%2Fr+=+-7%2F%28p%5E2%2B2%29
Computing for r:
if highlight%28p=1%29+
+r+%2B+1%2Fr+=+-7%2F%281%5E2%2B2%29
+%28r%5E2+%2B+1%29%2Fr+=+-7%2F3
+3%28r%5E2+%2B+1%29+=+-7r
+3r%5E2+%2B+3=+-7r
+3r%5E2+%2B7r%2B+3=+0
r+=+%281%2F6%29+%28sqrt%2813%29-7%29 or r+=+%281%2F6%29+%28-7-sqrt%2813%29%29
where %281%2F6%29+%28sqrt%2813%29-7%29+=+1%2F%28%281%2F6%29+%28-7-sqrt%2813%29%29%29-roots are reciprocal to each other
Thus, the actual roots are:
r%5B1+%5D=+%281%2F6%29+%28sqrt%2813%29-7%29+ or
r%5B2+%5D=+%281%2F6+%29%28-7-sqrt%2813%29%29
second solution:
if highlight%28p=2%29+
+r+%2B+1%2Fr+=+-7%2F%282%5E2%2B2%29
+%28r%5E2+%2B+1%29%2Fr+=+-7%2F6
+6%28r%5E2+%2B+1%29+=+-7r
+6r%5E2+%2B+6=+-7r
+6r%5E2+%2B7r%2B+6=+0...solving this using quadratic formula you will get

r+%5B1%5D=+-%281%2F12%29%2Ai%28sqrt%2895%29-7i%29 or r%5B2%5D+=+%281%2F12%29%2Ai+%28sqrt%2895%29%2B7i%29
where r+%5B1%5D=1%2Fr+%5B2%5D=> -%281%2F12%29%2Ai+%28sqrt%2895%29-7i%29=1%2F%28%281%2F12%29%2Ai+%28sqrt%2895%29%2B7i%29%29+
check our function: if highlight%28p=1%29+
%28p%5E2%2B2%29x%5E2%2B7x%2B3p=0
%281%5E2%2B2%29x%5E2%2B7x%2B3%2A1=0
3x%5E2%2B7x%2B3=0
roots are: x+=+%281%2F6%29+%28sqrt%2813%29-7%29 or x+=+%281%2F6%29+%28-7-sqrt%2813%29%29 which are reciprocal to each other

and, if highlight%28p=2%29+
%28p%5E2%2B2%29x%5E2%2B7x%2B3p=0
%282%5E2%2B2%29x%5E2%2B7x%2B3%2A2=0
6x%5E2%2B7x%2B6=0
roots are: => x=-%281%2F12%29%2Ai+%28sqrt%2895%29-7i%29 or x=%28%281%2F12%29%2Ai+%28sqrt%2895%29%2B7i%29%29+ and they are reciprocal to each other