SOLUTION: Let x-2,2√ 3, x-3 be the first 3 terms of a geometric sequence. Find the value of x (the quadratic equation can be solved by factoring).

Algebra ->  Radicals -> SOLUTION: Let x-2,2√ 3, x-3 be the first 3 terms of a geometric sequence. Find the value of x (the quadratic equation can be solved by factoring).      Log On


   

Question 986739: Let x-2,2√ 3, x-3 be the first 3 terms of a geometric sequence. Find the value of x (the quadratic equation can be solved by factoring).
Answer by ikleyn(52777) About Me  (Show Source):
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Since  x-2,  2sqrt%283%29 and  x-3%29  are the first  3  terms of a geometric sequence,  you have a proportion

2sqrt%283%29%2F%28x-2%29 = %28x-3%29%2F%282sqrt%283%29%29

(saying that the ratio of the second term to the first one is equal to the ratio of the third term to the second one).

From the proportion,

(x-2)*(x-3) = %282sqrt%283%29%29%5E2 = 4*3 = 12.

Hence,

x%5E2+-+2x+-+3x+%2B+6 = 12,

x%5E2+-+5x+-+6 = 0,

(x+1)*(x-6) = 0.

The roots are  -1  and  6.

Answer.  x = -1  or  x = 6.