Product of Years

Algebra Level 5

$\large (x + 2010)(x + 2011)(x + 2012)(x + 2013) = 5$

If the maximum real value of $x$ such that the above equation is true can be expressed in simplest terms as

$\large \frac{\sqrt{a+ b\sqrt{c}} -d}{e}$

For positive integers $a,b,c,d,e$ with $c$ square-free, and $\gcd (a,e) = 1$ .

Find $a+b+c-d +e$ .

This problem is not original.

The answer is -4006.

2 solutions

Samuel Bodansky
Feb 23, 2015

Same method, except for a slight difference in the substitution :)

Ayan Jain - 6 years, 3 months ago

Learn Latex and encode it

Francis Dave Cabanting - 3 years, 4 months ago

Terrell Bombb
Nov 23, 2016

let a = 2010, (x+a)(x+a+1)(x+a+2)(x+a+3)=5

expanding gets us: (x+a)^4 + 6(x+a)^3 + 11(x+a)^2 + 6(x+a) = 5

let y = x+a, then depressing the quartic will help solve it. but since we are living in the future and not the renaissance, we have the luxury of wolfram alpha. simply input: y^4 + 6y^3 + 11y^2 + 6y - 5

we get 2 real roots and 2 roots with imaginary dimensions. find the roots that fits the description of the problem then equate the root with x + 2010 to find x. substitute the value before doing the arithmetic operations.

Product of Years

The answer is -4006.

2 solutions

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