Challenge: Algebra Problem?

Question

Find a value of x satisfying the equation

5[sqrt(1-x) + sqrt(1+x)] = 6x + 8[sqrt(1 - (x^2))].

Bimeateater: Wow! How did you find that beastly factorization? Also, 2 of those solutions are extraneous.

It's mostly the rational solution I'm interested in since the problem doesn't explicitly ask for all solutions.

By the way this problem was used for training the 1996 USA team for the IMO (International Math Olympiad).

There is an alternative solution. Hint: Consider the range of values x can take on keeping in mind the constraints placed on it by the square root expressions. What does this suggest we might try substituting for x?

Answer 1

x = 0.96
x = (approximately) –0.99556, 0.5796, and 0.416

You end up with:

(25x - 24) * (100x^3 - 75x + 24) = 0

(That's why the x = 0.96 is exact.) The cubic equation is irreducible, but you can see the roots will lie between about -1 and (3/8)^0.5 and use that to seek approximate answers. Then refine them to some extent (i.e.: to make the cubic very close to zero) and you've got the next best thing to an answer.


Added:

Basically, I got lucky with the factoring. (Thank the US government's savings bond program back in 1969. That's where I learned 3/4 of $25 is $18.75. Learned it one 10¢ stamp at a time!)

You have 2500x^4 - 2400x^3 - 1875x^2 + 2400x - 576. Given the 2500 and 1875, a factor of 25x^? is a natural to try. That leads to a "leftover" of 600x - 576 and -2400x^3, both of which suggest 24 (25*24=600). A couple tries gives the factorization. Nothing easy comes a winner with the cubic factor so I entered it into a spreadsheet and narrowed down its three factors. Their values suggested nothing helpful for factoring and getting more exact results, so I left it at that.

Good point folks, about extraneous roots. I got swept up with the roots for the cubic factor and forgot about checking them back into the original equation.

But to answer a later answerer's assertion about 'you can take it as 0.96', well, no, it's exact.

So that leaves exactly 0.96 and approximately 0.416 which can be made accurate to more digits if one wishes.

Answer 2

Squaring both sides twice you end up with:

10000x^4 - 9600x^3 - 7500x^2 + 9600x - 2304 = 0
(which I believe another answerer got)

This is a beastly equation to solve normally, so I graphed it on GCalc and got a straight line! (Unbelievable)

The line meets the x - axis at (0.95652175, 0)
x = 0.95652175 = 0.96 (Approx) is one solution.

This is a biquadratic, so I know that the polar opposite will also be a root
x = -0.95652175 is also a solution.

But the graph does not intersect the x - axis at that point! So, it is an extraneous root.

It is a biquadratic, so it must have 4 roots, but as you pointed out, 2 or 3 will be extraneous as the equations was obtained by squaring the original one twice.

So, x = 0.95652175 is the only solution. You can take that to be 0.96

Answer 3

5[sqrt(1-x) + sqrt(1+x)] = 6x + 8[sqrt(1 - (x^2))].
5sqrt(1-x) + 5sqrt(1+x) = 6x + 8sqrt[(1-x)(1+x)]

squaring both sides you'll get
28x^2 - 14 = (96x - 50)√(1-x^2)

squaring again,
784x^4 - 784x^2 + 196
= (1-x^2)[9216x^2 - 9600x + 2500]

x^4(9216+784) - 9600x^3 + x^2(-784-9216+2500) + 9600x + (196-2500) = 0

10000x^4 - 9600x^3 - 7500x^2 + 9600x - 2304 = 0

(100x - 96)(100x^3 - 75x + 24) = 0

1 of them is x = 0.96

Answer 4

5[sqrt(1-x) + sqrt(1+x)] = 6x + 8[sqrt(1 - (x^2))]
5sqrt(1-x) + 5sqrt(1+x) = 6x + 8sqrt(1 - (x^2))
square bothe sides
25(1-x) + 50sqrt(1+x)sqrt(1-x)+25(1+x) = 36x^2 + 96xsqrt(1 - (x^2))+64(1-x^2)
50 + 50sqrt(1+x)(1-x) = 64-28x^2 + 96xsqrt(1 +x)(1-x)
28x^2-14 = 96xsqrt(1 +x)(1-x) - 50sqrt(1+x)(1-x)
I've messed up somewhere!

Answer 5

It sounds like 5x continues to be consistent and also you increment the multiplier on y by ability of two each and anytime period, with the first time period putting out at 0. regular time period: 5x + 2(t-a million)y the position "t" is the style of the time period. So for the first time period you get 5x because 2(a million-a million)y = 0. The fourth time period is 5x + 2(4-a million)y which simplifies to 5x + 6y. The fifth time period is 5x + 2(5-a million)y which simplifies to 5x +8y. The sixth time period is 5x + 2(6-a million)y which simplfies to 5x + 10y.

Answer 6

-8sqrt(1-x^2)+5sqrt(x+1)-6x+
5sqrt(1-x) =0

x =0.4159623 = 0.416 (approximately)

Answer 7

Challenge: do ur own homework