binomial expansion at a level?

Favorite Answer

1/sqrt[1-2x] - sqrt[1+2x] = ax^2 + bx^3

(1-sqrt[1-4x^2]) / sqrt[1-2x] = ax^2 + bx^3 {square both sides}

[1-2sqrt[1-4x^2] + 1-4x^2] / 1-2x = a^2x^4 + 2abx^5 + b^2x^6

1-2sqrt[1-4x^2] + 1-4x^2 = a^2x^4 + 2abx^5 + b^2x^6 - [2a^2x^5 + 4abx^6 + 2b^2x^7]

1-2sqrt[1-4x^2] + 1-4x^2 = a^2x^4 + (2ab-2a^2)x^5 + (b^2-4ab)x^6 - 2b^2x^7 {now isolate the radical}

-2sqrt[1-4x^2] = 2 - 4x^2 + a^2x^4 + (2ab-2a^2)x^5 + (b^2-4ab)x^6 - 2b^2x^7

Now you need to square both sides, which for obvious reasons I'm not going to do.

My guess is they want you to ignore all powers of x greater than 3 at each stage, but that seems pretty hokey to me.

Making this

-2sqrt[1-4x^2] = 2 - 4x^2 {then square}

4*(1-4x^2) = 2-16x^2 + 16x^4 {once again ignore the x^4}

4-16x^2 = 2-16x^2

4 = 2 which is a contradiction, so if the method of ignoring powers of x greater than 3 is true, there is no solution that satisfies this problem....Show more