If x 2 </msup> &#x2212;<!-- − --> 10 a x &#x2212;<!-- − --> 11 b = 0

Jerry Villegas

Jerry Villegas

Answered question

2022-05-22

If x 2 10 a x 11 b = 0 has roots c and d and x 2 10 c x 11 d = 0 has roots a and b, then find a + b + c + d ?

Answer & Explanation

pralkammj

pralkammj

Beginner2022-05-23Added 7 answers

In your approach, after doing componendo alone, you substitute a + b = a c, which should 10 c and it shouldn't give you anything new. Instead, we can do as follows which is straightforward calculation.
d = 10 a c and b = 10 c a
( a b ) ( c d ) = ( 11 b ) ( 11 d ) = 121 b d
If b = 0, then d = 0 = a = c and same for d = 0 a = 0 and c = 0 cases
Otherwise, a c = 121
a b = 10 a c a 2 = 11 d = 110 a + 11 c
a 2 110 a + 11 121 a 1210 = 0
a 3 10 11 a 2 10 11 2 a + 11 3 = 0
( a 3 + 11 3 ) 110 a ( a + 11 ) = 0
( a + 11 ) ( a 2 121 a + 11 2 ) = 0
a has 3 solutions and find solution set for each case c = 121 a

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