FA6, HA6 - Identities and algebraic arguments

FA6, HA6 - Identities and algebraic arguments

Learn the difference between an equation and an identity, and use equivalent expressions to explain why a calculation works for every input. This develops FA6 and HA6 through expansion, reverse expansion and short algebraic arguments about proposed rules.

Equations and identities

A number trick says: choose a number, add 44, double the result, then subtract twice your original number. Can it guarantee an answer of 88? First distinguish an equality that works for one input from one that works for every input.

The statement x+3=8x+3=8 is true at x=5x=5, but false at x=2x=2. In contrast, 2(x+3)=2x+62(x+3)=2x+6 is true for every real value of xx, because distributing the 22 produces exactly the expression on the right.

Equation and identity

An equation states that two expressions are equal. It can be true for some values, no values, or all permitted values of its variables. An identity is an equation true for all permitted values. The symbol ≡\equiv can emphasise an identity, as in 2(x+3)≡2x+62(x+3)\equiv2x+6.

“Permitted” refers to any stated restrictions: if a problem says nn is an integer, its claim concerns integers. The polynomial expressions in this lesson are defined for every real input.

StatementWhen is it true?Identity?
x+3=8x+3=8Only at x=5x=5No
2(x+3)=2x+62(x+3)=2x+6At every real xxYes
x+3=x+4x+3=x+4At no real xxNo

Testing x=0x=0 in (x+2)2=x2+4(x+2)^2=x^2+4 gives 4=44=4, but testing x=1x=1 gives 9≠59\ne5. One successful input therefore does not establish an identity. One failed permitted input is enough to show that a proposed identity is false.

Key idea

An identity is a special kind of equation. The distinction concerns which values make it true, not whether its printed symbol is == or ≡\equiv.

Choose the equality that survives every real input.

Equivalence

Equivalent expressions have the same value for every permitted input. Expanding and factorising preserve the value, so they let you establish an identity without trying individual inputs.

To show two expressions are equivalent, start with one and use valid algebra to obtain the other. Alternatively, simplify both sides separately until they have the same terms. Do not assume the proposed identity is true as the first step of your argument.

Worked example 1: establishing an identity by expansion

Question. Show that (x+2)(x+5)≡x2+7x+10(x+2)(x+5)\equiv x^2+7x+10.

Plan. Expand the left side and collect its middle terms.

(x+2)(x+5)=x2+5x+2x+10multiply all four pairs=x2+7x+10collect like terms\begin{aligned} (x+2)(x+5)&=x^2+5x+2x+10 &&\text{multiply all four pairs}\\ &=x^2+7x+10 &&\text{collect like terms} \end{aligned}

Every step holds for any real xx.

Answer. The expressions are equivalent, so the statement is an identity.

Check. At x=1x=1, both give 1818. This checks the arithmetic at one input; the expansion supplies the general argument.

Change only +5+5 to −5-5 in the brackets. This time read the relationship backwards, turning the expanded expression into factors.

Worked example 2: establishing an identity by factorisation

Question. Show that x2−3x−10≡(x+2)(x−5)x^2-3x-10\equiv(x+2)(x-5).

Plan. Split −3x-3x into 2x−5x2x-5x, then reverse distribution twice.

x2−3x−10=x2+2x−5x−102x−5x=−3x=x(x+2)−5(x+2)factor each pair of terms=(x−5)(x+2)both terms contain (x+2)\begin{aligned} x^2-3x-10&=x^2+2x-5x-10 &&2x-5x=-3x\\ &=x(x+2)-5(x+2) &&\text{factor each pair of terms}\\ &=(x-5)(x+2) &&\text{both terms contain }(x+2) \end{aligned}

Why those split terms? The numbers 22 and −5-5 add to −3-3 and multiply to −10-10, making the same bracket appear in both groups.

Answer. x2−3x−10≡(x+2)(x−5)x^2-3x-10\equiv(x+2)(x-5), since bracket order does not change a product.

Check. Expanding the final product gives x2−5x+2x−10=x2−3x−10x^2-5x+2x-10=x^2-3x-10. At x=1x=1, both expressions give −12-12.

An identity can be used in either direction. Which direction is useful depends on the question: expansion compares terms; factorisation exposes a product.

Continue this lesson.

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