Expert: Josh - 2/1/2009

Question
(-6)         -3
-------    ------
(3x-2)^2 =    2

How do i Solve for x?

Answer
Mike,

The problem we have here is that the unknown "x" sits at the bottom of a fraction. To overcome this, we use the cross-multiplication technique to get rid of the fractions. I'll explain what I mean by this in due course.

The basis for this is that we can perform the same operation on both sides of the equation without changing the truth in the expression.
i.e., we can multiply both sides by 2(3x-2)^2, for instance, to get a "LEFT HAND SIDE = RIGHT HAND SIDE" expression without fractions while everything still holds. Carrying out this step, we get [-6*2(3x-2)^2]/(3x-2)^2 = [-3*2(3x-2)^2]/2.

Notice that there will always be common factors on either side. So,
[-6*2(3x-2)^2]/(3x-2)^2 = [-3*2(3x-2)^2]/2 simplifies to
[-6*2]/1 = [-3(3x-2)^2]

As promised, we have effectively multiplied the numerator on each side of the equation by the denominator on the opposite side of the equation. Basically, we reduce A/B = C/D to A*D = B*C.

This gives -6*2 = -3(3x-2)^2. From this point on, you can expand the RHS, gather terms into a quadratic polynomial. Use either factorization or the quadratic equation x=[-b+-sqrt(b^2-4ac)]/(2a) to obtain the solution(s).

-12 = -3*(9x^2-12x+4)
4   = 9x^2-12x+4
3x(3x-4)=0 when either 3x=0 or 3x-4=0

solutions: x=0 or x=4/3

Let me know if you need further explanation.