SAT Math · Equivalent expressions · medium

For which values of x is the expression (x²-4)/(x²-2x) undefined?

For which values of xx is the expression x2−4x2−2x\dfrac{x^2-4}{x^2-2x} undefined?

Answer and explanation

Answer: D: x=0 and x=2

An expression is undefined where its denominator equals zero. Factor the denominator: x2−2x=x(x−2)x^2 - 2x = x(x - 2), which is zero when x=0x = 0 or x=2x = 2. That makes the answer D. Although the numerator x2−4=(x−2)(x+2)x^2 - 4 = (x-2)(x+2) shares the factor x−2x - 2 with the denominator, canceling would hide the problem at x=2x = 2, but the original expression is still undefined there. Choice A misses x=0x = 0. Choice B uses the zeros of the numerator, ±2\pm 2, which are where the expression equals zero (or is a hole) and not where it is undefined. Choice C misses x=2x = 2.

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