Determine the quadratic ax^2+bx+c, if a,b,c are terms of arithmetic sequence. a=2t-3,b=5t+1,c=4t-7

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We are given that in the quadratic equation ax^2+bx+c, a,b,c are terms of arithmetic sequence such that a = 2t-3, b = 5t+1 and c = 4t-7

As consecutive terms of an AP have a common difference:

4t - 7 - 5t - 1 = 5t + 1 - 2t + 3

=> -t - 8 = 3t + 4

=> 4t = -12

=> t  = -3

a = 2t - 3 = -6 - 3 = -9

b = 5t + 1 = -15 + 1 = -14

c = 4t - 7 = -12 - 7 = -19

The quadratic equation is -9x^2 - 14x - 19 = 0

=> 9x^2 + 14x + 19 =  0

The required quadratic equation is 9x^2 + 14x + 19 =  0

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