Question 825596: If t is an unknown constant, which binomial must be a factor of 7m^2 + 14m - tm -2t?
Answer by Fermat(136)   (Show Source):
You can put this solution on YOUR website! 7m^2 + 14m - tm - 2t
Assume that the above quadratic is factorable.
let there be two factors of the term -2t, call them a and b, then
(7m + a)(m + b) = 7m^2 + (14 - t)m - 2t
7m^2 + (7b + a)m + ab = 7m^2 + (14 - t)m - 2t
equating coefficients,
7b + a = 14 - t
ab = -2t
substituting for a = 14 - t - 7b into the 2nd equation,
7b^2 - (14 - t)b - 2t = 0
(7b + t)(b - 2) = 0
b = -t/7, (a = 14)
or
b = 2, (a = -t)
If a = -t, b = 2, then
7m^2 + (14-t)m -2t = (7m - t)(m + 2)
Then the binomials (7m - t) and (m + 2) are factors of the original quadratic expression.
If a = 14, b = -t/7, then
7m^2 + (14-t)m -2t = (7m + 14)(m - t/7)
Then the binomials (7m + 14) and (m - t/7) are factors of the original quadratic expression.
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