Asked by Bob
The expression $x^2 + 3x - 28$ can be written as $(x + a)(x - b),$ and the expression $x^2 - 10x - 56$ written as $(x + 2b)(x + c)$, where $a$, $b$, and $c$ are integers such that $c > 0.$ What is the value of $2c - a$?
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Answered by
GPT-5 mini
AI
From (x+a)(x-b) = x^2 + (a-b)x - ab = x^2 + 3x - 28, we have
a - b = 3 and ab = 28. Solving gives (a,b) = (7,4) or (−4,−7).
Use (x + 2b)(x + c) = x^2 + (2b + c)x + 2bc = x^2 − 10x − 56.
If b = 4 then 2b + c = 8 + c = −10 ⇒ c = −18 (not > 0), so reject.
If b = −7 then 2b + c = −14 + c = −10 ⇒ c = 4 (and 2b·c = −14·4 = −56, consistent).
Thus a = −4 and c = 4, so 2c − a = 8 − (−4) = 12.
a - b = 3 and ab = 28. Solving gives (a,b) = (7,4) or (−4,−7).
Use (x + 2b)(x + c) = x^2 + (2b + c)x + 2bc = x^2 − 10x − 56.
If b = 4 then 2b + c = 8 + c = −10 ⇒ c = −18 (not > 0), so reject.
If b = −7 then 2b + c = −14 + c = −10 ⇒ c = 4 (and 2b·c = −14·4 = −56, consistent).
Thus a = −4 and c = 4, so 2c − a = 8 − (−4) = 12.
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