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Let f(x) = x^2 + bx + 12 - 3x + 10 for all real numbers $x$. Find the greatest integer value of $b$ such that $-4$ is not in the range of $f(x)$.

 Jan 19, 2025
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x^2 + (b-3)x + 22  > -4

 

x^2 + (b-3)x + 26 >  0

 

Set the discriminant to < 0 

 

(b - 3)^2  - 4(1)(26) < 0

 

(b -3)^2 < 104

 

b -3  <   [ sqrt 104]

 

b- 3 < ≈ 10.2

 

b <  ≈13.2  

 

Greatest  integer value of b = 13

 

cool cool cool

 Jan 19, 2025

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