Chapter 1 · Algebraic Proofs · Concept 4
1-10 Proving Equalities (1)
In lesson 1-9 you were given an identity and asked to find the constants. Now you are given the whole thing and asked to show it is true.
Guiding question
How do you show two expressions are equal without assuming they already are?
Learning outcome. Prove the validity of algebraic equalities and identities in regular cases (without any prerequisites).
The whole method
Simplify the left side and the right side separately, until they become the same expression.
Often only one side needs any work. Sometimes both do. Either way, the two sides are tidied on their own and the proof is finished when they match.
Never move anything across the equals sign.
When you multiply both sides by something, or add the same thing to both sides, you are treating the statement as if it were already true. But being true is exactly what you were asked to show. A proof that starts by assuming its own answer proves nothing — even if every line after it is correct.
Not this
(x+y)² − (x−y)² = 4xy
→ (x+y)² = 4xy + (x−y)²
You have used the thing you are proving.
This
LHS = (x+y)² − (x−y)²
= … = 4xy = RHS
Each side is tidied alone. Nothing was assumed.
This is how a proof is written out. The two columns never touch — there is nowhere for anything to cross to.
Prove: (x + y)² − (x − y)² = 4xy
LHS
- (x + y)² − (x − y)²
- …
RHS
- 4xy
Neither column ever touches the other. Nothing crosses the equals sign.
2. Prove that (a2 + b2)(c2 + d2) = (ac − bd)2 + (ad + bc)2
- 1
Take the left side on its own.
LHS = (a2 + b2)(c2 + d2)Always label it. It makes clear you are working on one side only.
Work with a partner. Do these together, then talk about the answers. Write LHS = … and RHS = … in two separate places on the page.
Prove the following equalities.
Prove that (a2 − ab + b2)(a2 + ab + b2) = (a2 + b2)2 − (ab)2
Never work on both sides at once.
Multiplying both sides, or moving a term across, assumes the statement is already true. That is the thing you were asked to prove, so the proof is worth nothing.
Label LHS and RHS.
Write "LHS =" and "RHS =" and keep them apart on the page. It is how the examiner sees that you never mixed them, and it stops you doing it by accident.
Finish with a sentence.
End with "and so the two sides are equal". Working that stops without saying what it showed is not a proof yet.
Look for a grouping before expanding.
(x²+1+x)(x²+1−x) is a difference of two squares. Spotting that turns nine products into two, and every product you avoid is a slip you cannot make.
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