📖 The Lesson
An expression is called undefined when the math situation is impossible — it simply cannot exist in normal real-number math. On the GED, there are two main ways an expression becomes undefined:
√(negative) = undefined
No real number multiplied by itself gives a negative result, so the square root of a negative has no real-number answer.
any0 = undefined
You can't divide something into zero groups. The numerator can be anything — only the denominator decides if the fraction is undefined.
√ Part 1 — Square Roots of Negative Numbers
Think it through. You know that 2 × 2 = 4 and (−2) × (−2) = 4, so √4 = 2. Both work fine. Now ask: what number multiplied by itself equals −4? No matter what real number you try, multiplying it by itself always gives a positive result. There is no real number whose square is negative.
The Rule
If the number inside the square root is positive → defined
If the number inside is zero → defined (√0 = 0)
If the number inside is negative → UNDEFINED
Always simplify inside the root first. If the result is negative, the expression is undefined.
√(−9)
Undefined
Negative inside the root
√(−25)
Undefined
Negative inside the root
√(2 − 8)
Undefined
2 − 8 = −6 → negative
√16
Defined
16 is positive
√(3 + 6)
Defined
3 + 6 = 9 → positive
No-calculator tip: Always simplify inside the root first. The moment you see a negative result inside the bars, the expression is undefined — no further work needed.
x0 Part 2 — Fractions with Zero on the Bottom
A fraction means part out of total. 24 means 2 parts out of 4 — that makes sense. 05 means 0 parts out of 5 — that just equals 0. But 20 means 2 parts out of zero total? That's impossible.
The Rule
If the denominator is zero → UNDEFINED
If the denominator is anything else → defined (even if the numerator is 0)
Very important — ignore the top! When checking for undefined fractions, only the denominator matters. The numerator can be zero, negative, or huge — it does not make the expression undefined. Only the bottom controls undefined.
50
Undefined
Denominator = 0
x0
Undefined
Bottom is always 0
05
Defined
0 on top → answer is 0
−84
Defined
Negative answer is fine
🎯 Part 3 — When Variables Are Involved
Most GED questions ask: "For what value of x is this expression undefined?" The answer: find the value of x that makes the denominator zero.
Worked Example — Linear Denominator
For what value of x is x + 3x − 4 undefined?
Ignore the top. Set the bottom equal to 0: x − 4 = 0
Solve: x = 4
Undefined at x = 4
Worked Example — Quadratic Denominator
For what values of x is 2x − 5x² − 16 undefined?
Set the bottom equal to 0: x² − 16 = 0
Factor: (x − 4)(x + 4) = 0
x = 4 or x = −4
Undefined at x = 4 and x = −4
GED Tip — Numerator trap: The GED sometimes tries to trick you by making the numerator zero. Don't fall for it. In (x − 4)(x + 1)x + 2, plugging in x = 4 gives 06 = 0 — defined! Only x = −2 makes the bottom zero, so x = −2 is the value that makes the expression undefined.
📝 Summary
An expression is undefined when (1) a square root contains a negative number, or (2) a fraction has zero in the denominator. Always simplify inside the root before judging it. For fractions, ignore the numerator entirely — only the denominator decides. When variables are involved, set the denominator equal to zero and solve. Those values of x are where the expression is undefined.
Two patterns to memorize: √(negative) = undefined and any0 = undefined. Master these two and you'll handle every undefined expression question on the GED.