Recognizing Equivalent Expressions

Equal at every x, or a near-miss that is not.

Equal at every x

Two expressions are equivalent when they give the same value for every value of the letter. 3(x + 2) and 3x + 6 are equivalent: at x = 1 both are 9, at x = 10 both are 36, and there is no value of x where they differ.

Expanding, factoring and collecting like terms all turn an expression into an equivalent one. They change how it is written, never its value. That is why they are allowed.

3x6x233(x + 2) = 3x + 6

A rectangle 3 tall and x + 2 long. It is made of a 3 by x piece and a 3 by 2 piece, so its area is 3x + 6.

The near-miss

3x + 2 looks close to 3(x + 2), but it is not equivalent. The 3 multiplies everything inside the bracket, so the 2 becomes 3 × 2 = 6. In 3x + 2 the 2 was never multiplied by 3.

One value is enough to show it. At x = 1, 3(x + 2) = 3 × 3 = 9, but 3x + 2 = 5. The two expressions disagree at x = 1, so they are not equivalent.

Squaring a sum

Another near-miss is (x + 3)² = x² + 9. Write the square as two brackets and multiply out: (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9. The middle term, 6x, is real.

At x = 1, (x + 3)² = 16 and x² + 6x + 9 = 16, but x² + 9 = 10. So x² + 9 is not equivalent to (x + 3)².

x²3x3x9x3x3x² + 6x + 9

A square of side x + 3 holds x², 9, and two strips of 3x. Writing x² + 9 leaves out both strips.

One match proves nothing

Agreeing at one value is not enough. x² and 2x are both 4 when x = 2. But at x = 3, x² = 9 and 2x = 6, so they are not equivalent. Two different expressions can meet at a value or two and still be different everywhere else.

So substitution works in only one direction. One value where two expressions disagree proves they are not equivalent. Values where they agree prove nothing, however many you try. To show that two expressions are equivalent, use algebra: expand or simplify both until they are written the same way.

−10123456xx²2xx² = 1, 2x = 2x = 1: ≠5x + 4 = 2x + 19x² = 2x ?

x = 1: x² = 1 and 2x = 2, different, so x² and 2x are not equivalent, whatever happened at x = 2; a disagreement at one x is a proof, an agreement at one x is not

Find every x where the two sides agree

The curve is x² and the straight line is 2x. Slide x and read both values. They agree at x = 0 and at x = 2, and nowhere else.

Worked example: Three Pupils' Expressions for the Tiles Round a Square Pond

Question A square pond is n tiles long on each side, and one row of square tiles is laid all the way round it. Aisha says the border needs (n + 2)2 − n2 tiles. Ben says it needs 4(n + 1) tiles. (a) Show that the two expressions are equivalent. (b) Chen's expression is n2 + 8. It gives 12 tiles when n = 2, and so do the other two. Is Chen's expression equivalent to them?

  1. 1.Aisha takes the pond, n2 tiles, away from the large square of side n + 2. Expand: (n + 2)2 = n2 + 4n + 4, so (n + 2)2 − n2 = 4n + 4.

    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tiles(n + 2)2− n2= 4n + 4
    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tiles(n + 2)2− n2= 4n + 4
    Aisha: the large square minus the pond. (n + 2)2 − n2 = n2 + 4n + 4 − n2 = 4n + 4.

  2. 2.Ben counts four runs of n + 1 tiles, each run being one side and one corner. Expand: 4(n + 1) = 4n + 4.

    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tiles(n + 2)2− n2= 4n + 44(n + 1) = 4n + 4
    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tiles(n + 2)2− n2= 4n + 44(n + 1) = 4n + 4
    Ben: four runs of n + 1 tiles. 4(n + 1) = 4n + 4.

  3. 3.(a) Both expressions simplify to 4n + 4, so they are equal for every value of n. They are equivalent.

    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tiles(n + 2)2− n2= 4n + 44(n + 1) = 4n + 4both are 4n + 4: equivalent
    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tiles(n + 2)2− n2= 4n + 44(n + 1) = 4n + 4both are 4n + 4: equivalent
    (a) Both expressions simplify to 4n + 4, so they are equivalent.

  4. 4.When n = 2, 4n + 4 = 12 and n2 + 8 = 4 + 8 = 12. One value that agrees does not prove that two expressions are equivalent, because two different expressions can be equal at a single value.

    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tilesn4n + 4n2+ 821212
    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tilesn4n + 4n2+ 821212
    When n = 2 both 4n + 4 and n2 + 8 are 12. One value that agrees is not a proof.

  5. 5.Test another value. When n = 3, 4n + 4 = 16 but n2 + 8 = 9 + 8 = 17.

    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tilesn4n + 4n2+ 82121231617
    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tilesn4n + 4n2+ 82121231617
    When n = 3, 4n + 4 = 16 but n2 + 8 = 17.

  6. 6.(b) Chen's expression is not equivalent to the others: when n = 3 it gives 17 tiles and the border needs 16. One value that disagrees is enough to show this.

    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tilesn4n + 4n2+ 82121231617n2+ 8 is not equivalent to 4n + 4
    n + 2 tiles on each sidepondn by ndrawn for n = 3: 16 border tilesn4n + 4n2+ 82121231617n2+ 8 is not equivalent to 4n + 4
    (b) Chen's expression is not equivalent: when n = 3 it gives 17 tiles and the border needs 16.

Answer: (a) Both simplify to 4n + 4; (b) No: when n = 3 Chen's expression gives 17 tiles and the border needs 16

Common mistakes

  • Deciding that n2 + 8 is equivalent because it agrees when n = 2. Two different expressions can be equal at one value. Equivalent expressions are equal at every value.
  • Expanding (n + 2)2 as n2 + 4. (n + 2)2 = (n + 2)(n + 2) = n2 + 2n + 2n + 4, so the 4n is part of it.

More algebraic expressions problems, worked step by step →

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