We need to solve the equation without performing an operation that removes a possible case. We will factor first, solve each factor separately, check every candidate in the original equation and only then state the complete solution set.
Step 1: factor out the common variable
Both terms in the quadratic equation x2−5x=0 share a common variable factor x. Decompose each term and factor x outside the brackets:
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x2−5x=0(Given quadratic equation)
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x⋅x−5⋅x=0(Identify common factor x)
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x(x−5)=0(Factor out x)
Step 2: apply the zero-product rule
By the Zero-Product Property, if the product of two real expressions equals zero (A⋅B=0), then at least one factor must equal zero (A=0 or B=0).
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x(x−5)=0
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Applying Zero-Product Property: x=0orx−5=0
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Case 1: x=0
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Case 2: x−5=0⟹x−5+5=0+5⟹x=5
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Combined solution set: x=0orx=5
Step 3: check both candidates in the original equation
Check the first candidate (x=0):
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LHS=(0)2−5(0)
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LHS=0−0=0
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LHS=RHS=0(Valid root ✓)
Check the second candidate (x=5):
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LHS=(5)2−5(5)
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LHS=25−25=0
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LHS=RHS=0(Valid root ✓)
Both substitutions evaluate to true identity statements (0=0), confirming that the equation has two distinct real roots.
Step 4: identify what division by x assumes
Here is the incorrect method that students frequently attempt:
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x2−5x=0
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Dividing both sides by x implicitly assumes x=0
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xx2−5x=x0
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xx2−x5x=0
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x−5=0⟹x=5
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Warning: The root x=0 was discarded because dividing by zero is undefined!
The reusable rule
If every term contains a variable factor, factor it out first.
Never cancel a variable that could equal zero unless the zero case has already been considered.
Check the number of roots. A quadratic can have two, one or no real roots; an operation must not change its valid solution set.
Substitute answers into the original equation, not only into a later equation that may have lost a case.
Final step: state every verified root
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x=0orx=5
Young Math Brains asks what a cancellation assumes before allowing it, then verifies every root in the original quadratic equation.
