A number theory problem by abhishek anand
Number Theory
Level
3
a^3+b^3=10; a^2+b^2=7; a+b=x; find the gretest value of x; hint: it is a number between -4 to10
-5
-1
4
5
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1 solution
This problem can be approached by elementary factoring.
a^3 + b^3 = 10 = (a+b)(a^2-ab+b^2)
From here, we can plug in the given expression "a^2 + b^2 = 7" and "a + b = x" into the factored expression above. Now, we have:
(x)(7 - ab) = 10
To replace "ab" with some form of x, we can square the expression (a+b) = x so that it is now, then we can modify it algebraically.
a^2 + 2ab + b^2 = x^2 7 + 2ab = x^2 2ab = x^2 - 7 ab = (x^2-7) * (1/2)
We plug this into our previous expression: (x)(7 - ab) = 10 (x)(7 - (x^2-7) * (1/2)) = 10 7x - (x^3-7x) * (1/2) = 10 14x - (x^3-7x) = 20 14x - x^3 + 7x = 20 x^3 - 21x + 20 = 0
Visually, we can see that x = 1 is a simply solution. With that, we can factor the rest by using either long division or synthetic division. Finally we end up with: (x-1)(x+5)(x-4) = 0 The greatest value of x is 4.
x=4