Family Functions Related to Phi

Algebra Level 3

Let $f_n(x)$ be a family of quadratics with integer coefficients in the form $ax^2+bx+c$ such that at least one of its roots is $\phi$ , which is $\frac{1+\sqrt{5}}{2}$ . There exists a quadratic in the family $f_n(x)$ such that the sum $a+b+c=-1$ . The name of this particular function is $f_{\star}(x)$ . Find $f_{\star}(16)$ .

The answer is 239.

1 solution

Daniel Liu
Apr 12, 2014

Note that the second root must be the conjugate of the first root, or $\dfrac{1-\sqrt{5}}{2}$ . Thus, the quadratic is $\left(x-\dfrac{1+\sqrt{5}}{2}\right)\left(x-\dfrac{1-\sqrt{5}}{2}\right)=x^2-x-1$ . Since this quadratic fits the condition that $a+b+c=-1$ , then it is our desired quadratic, and so our answer is $256-16-1=\boxed{239}$ .

*Note: * Even if $a+b+c=n$ for any $n$ , we can still solve for the quadratic by multiplying the entire thing by a constant such that $a+b+c=n$ .

This is how I solved it perfectly. Why did you have to beat me to the solution?

Sharky Kesa - 7 years, 2 months ago

Yes! Dude, he always beats me to the solution! Arg! :D

Finn Hulse - 7 years, 2 months ago

What cool about the family is that any quadratic in the form $ax^2-ax-a$ will have a root of $\phi$ ! So usually its just luck if you stumble upon the function the very first time. Check out some other uses for this equation here !

Finn Hulse - 7 years, 2 months ago

Family Functions Related to Phi

The answer is 239.

1 solution

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