The solution that is not one
Producing an extraneous root on purpose, then working out where it came from.
Scenario
Consider the equation
the square root of (x + 6) = x
where the square root sign means the non-negative root, as it always does on this test.
Work through it without skipping the check at the end.
Answers in order, explanations withheld, and a report on where the chain broke.
- 22.1RadicalsStandard
Squaring both sides gives which equation?
- Ax + 6 = x^2
- Bx + 6 = x
- Cx^2 + 36 = x^2
- Dx + 36 = x^2
Show the answer and reasoningHide the reasoning
x + 6 = x^2
Squaring the left side removes the radical and leaves what was under it; squaring the right side gives x^2. The 6 is inside the radical, so it is not squared separately.
- 22.2Quadratic equationsStandard
Builds on: The equation from question 1. Squaring the 6 instead would give a quadratic with no whole-number roots at all.
What are the solutions to that quadratic?
- A3 and -2
- B-3 and 2
- C6 and -1
- D3 and 2
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3 and -2
Rearranged: x^2 - x - 6 = 0, which factors as (x - 3)(x + 2). So the quadratic has solutions 3 and -2.
- 22.3RadicalsChallenge
Builds on: Question 2 produced two candidates. This is the step that a single question can be got right without ever performing, because a lone answer choice will usually only list one of them.
Which of those solutions satisfies the original equation?
- ABoth of them
- BOnly 3
- COnly -2
- DNeither of them
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Only 3
Test 3: the square root of 9 is 3, and the right side is 3. Test -2: the square root of 4 is 2, and the right side is -2, so the equation says 2 = -2, which is false.
- 22.4RadicalsChallenge
Builds on: Question 3 found the failure and this names it. Getting question 3 right by testing, without this, leaves a reader checking out of habit rather than because they know what they are checking for.
Why did the quadratic produce a solution the original equation does not have?
- ABecause a mistake was made in the factoring.
- BBecause squaring turns a false statement into a true one when the two sides differ only in sign.
- CBecause the original equation has no solutions.
- DBecause negative numbers cannot be square roots.
Show the answer and reasoningHide the reasoning
Because squaring turns a false statement into a true one when the two sides differ only in sign.
2 = -2 is false, but squaring both sides gives 4 = 4, which is true. Squaring can only add solutions, never lose them, so every root of the squared equation has to be tested against the original — and this is why the check is a step rather than a formality.