SOLUTION: Please help me solve this: Given {{{ f(x)= 2(x-3)(x+2)^2+2(x+2)(x-3)^3 }}} show that {{{ x^2-5x+11>0 }}} for all integers x.

Algebra ->  Algebra  -> Equations -> SOLUTION: Please help me solve this: Given {{{ f(x)= 2(x-3)(x+2)^2+2(x+2)(x-3)^3 }}} show that {{{ x^2-5x+11>0 }}} for all integers x.      Log On

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Question 516980: Please help me solve this:
Given +f%28x%29=+2%28x-3%29%28x%2B2%29%5E2%2B2%28x%2B2%29%28x-3%29%5E3+
show that +x%5E2-5x%2B11%3E0+ for all integers x.
Answer by richard1234(5390) About Me  (Show Source):
You can put this solution on YOUR website!
The "given" part is rather pointless, because we can prove that statement for all integers x without it.

The roots of the quadratic equation are:



This is a complex number, so the quadratic has no real roots. This means that the function is either all positive or all negative. Since the x^2 coefficient is positive, the entire function must be positive (since it opens upward and a quadratic having any negative values would be bounded by the x-axis).