The decision and its conditions
An algebraic transformation may produce candidates rather than equivalent solutions. Multiplying by a denominator requires its original nonzero condition; squaring a radical equation loses a sign condition. Preserve those restrictions and substitute candidates back into the original equation. A nonlinear system combines curves and may have more than one intersection. Exponential expressions with repeated powers can sometimes become a quadratic after a positive substitution such as u=2^x. The substituted variable has its own range, which must be checked before returning to x.
Worked example
Solve [(x−3)(x−6)]/(x−3)=0 over the reals. What is the allowed solution? The denominator excludes x=3. For allowed x, the equation simplifies to x−6=0. Thus x=6; the other numerator zero remains forbidden.
A contrasting decision
How many real solutions does √(x+1)=x−1 have? The right side must be nonnegative, so x≥1. Squaring gives candidates x=3 and x=0. Only 3 satisfies x≥1; direct substitution gives √4=2. There is one real solution.
Diagnose a competing method
The first example’s tempting answer 9 follows this mistake: Add the numerator zeros instead of selecting the allowed root. Exclude denominator zeros before applying the zero-numerator condition.
Transfer before revealing
The curves y=x² and y=4x meet at the origin and one other point. What is the y-coordinate of the other intersection? Pause and identify the original restrictions. Solution: Set x²=4x, giving x(x−4)=0. Besides the origin, the intersection has input 4. The output is y=4²=16; this also equals 4×4 in the line equation.
Check the boundary of the method
Exponential expressions with repeated powers can sometimes become a quadratic after a positive substitution such as u=2^x. The sign of the original right side survives as a restriction after squaring. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
Solve [(x−3)(x−6)]/(x−3)=0 over the reals. What is the allowed solution?
- The denominator excludes x=3.
- For allowed x, the equation simplifies to x−6=0.
- Thus x=6; the other numerator zero remains forbidden.
Compare the other answer choices
Confuse a function value of zero with an input of zero.
Add the numerator zeros instead of selecting the allowed root.
Reverse the sign when solving the remaining linear equation.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overview