If [(x+2)/(x-2)]-[(x-2)/(x+2)]=[(x-1)/(x+3)]-[(x+3)/(x-3)] then the values of x are? (a)0,+-√6 (b)0,+-√3 (c)0,+-2√3 (d)None The correct answer is d Please explain ?

Question

Answer

Solution:

Given equation is,

[(x 2)/(x-2)]-[(x-2)/(x 2)]=[(x-1)/(x 3)]-[(x 3)/(x-3)]

Multiplying both sides by (x-2)(x+2)(x+3), we get

(x+2)(x+3)x-(x-2)(x+3)(x+2)=(x-2)(x+2)-(x+3)(x 2)

Expanding the left-hand side and right-hand side, we get

x^3-2x^2-11x+12=-x^3+7x-18

Adding x^3 and 2x^2 to both sides, we get

2x^3-11x+12=7x-18

Bringing all the terms to one side, we get

2x^3-18x+30=0

Dividing both sides by 2, we get

x^3-9x+15=0

We can't find the values of x by factoring, so we will use the rational root theorem to check for rational roots.

The rational root theorem states that if the polynomial equation

a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0,

where all the coefficients a_n, a_{n-1}, ..., a_1, a_0 are integers, has a rational root p/q, where p and q are integers with no common factors other than 1 and q ≠ 0, then p must divide a_0 and q must divide a_n.

In our equation, a_n=1, a_{n-1}=0, a_{n-2}=-9, and a_0=15. The only possible rational roots are ±1, ±3, ±5, ±15. However, none of these roots work, so there are no rational roots.

Therefore, the values of x are not rational, so the answer is (d) none.

Answer

Similar CA Foundation Doubts

If x^1/p = y^1/q = z^1/r and xyz = 1, then the value of p q r is .?? Options are

If 2^(2x 3)-3^2×2^x 1=0 then the values of x are?

If 2^(2x 3)-3^2.2^x 1=0 then the values of x are?