2008 iTest Problems/Problem 85

Problem

Let $(a,b,c,d)$ be a solution to the system $\begin{align*}a+b&=15,\\ab+c+d&=78,\\ad+bc&=160,\\cd&=96.\end{align*}$ Find the greatest possible value of $a^2+b^2+c^2+d^2$ .

Solution

Note that when multiplying quadratics, terms add up similar to the equations of a system, so let $\begin{align*} p(x) &= (x^2 + ax + c)(x^2 + bx + d) \\ &= x^4 + (a+d)x^3 + (ab+c+d)x^2 + (ad+bc)x + cd \\ &= x^4 + 15x^3 + 78x + 160x + 96 \end{align*}$ Factoring $p(x)$ with the Rational Root Theorem results in $(x+4)(x+4)(x+1)(x+6)$ . By the Fundamental Theorem of Algebra, we know that $x+4, x+4, x+1, x+6$ are all the linear factors of the polynomial, so the quadratic factors can only be multiplied from these linear factors.

There are only two possible distinct groupings (not counting rearrangements) -- $(x^2 + 8x + 16)(x^2 + 7x + 6)$ and $(x^2 + 5x + 4)(x^2 + 10x + 24)$ . In the first case, $a^2 + b^2 + c^2 + d^2 = 405$ , and in the second case, $a^2 + b^2 + c^2 + d^2 = 717$ . The largest of the two options is $\boxed{717}$ .

2008 iTest (Problems)
Preceded by: Problem 84	Followed by: Problem 86
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2008 iTest Problems/Problem 85

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