Algebraic Identities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Algebraic Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Algebraic identities are equalities true for all permitted values of their variables.

Identities are always-true shortcuts — no matter what values you substitute, both sides will always be equal.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An algebraic identity is an equation that holds for all permitted values of its variables, so it acts as an always-valid rewriting shortcut.

Common stuck point: The procedure for algebraic identities is the easy part; the trap is dropping the middle term: writing (a+b)2=a2+b2(a+b)^2=a^2+b^2. Asking "Is this equality true for every value of the variable (an always-true pattern), rather than only for special values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this equality true for every value of the variable (an always-true pattern), rather than only for special values?

Worked Examples

Example 1

easy
Expand (x+3)2(x + 3)^2 using the identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2.

Answer

x2+6x+9x^2 + 6x + 9

First step

1
Step 1: a=xa = x, b=3b = 3.

Full solution

  1. 2
    Step 2: (x+3)2=x2+2(x)(3)+32=x2+6x+9(x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9.
  2. 3
    Check: (x+3)(x+3)=x2+3x+3x+9=x2+6x+9(x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 ✓
Algebraic identities are equations that hold for all values of the variables. The perfect square identity (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 is faster than FOIL for squaring binomials.

Example 2

medium
Compute 47247^2 mentally using (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 with a=50,b=3a = 50, b = 3.

Example 3

medium
Factor x4−16x^4 - 16 completely.

Example 4

hard
Simplify x2−4x2−4x+4\dfrac{x^2 - 4}{x^2 - 4x + 4}.

Example 5

challenge
Show that for any reals a,ba,b: a4+b4+(a+b)4=2(a2+ab+b2)2a^4 + b^4 + (a+b)^4 = 2(a^2 + ab + b^2)^2.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Expand (2x−5)2(2x - 5)^2.

Example 2

hard
Factor x3+8x^3 + 8 using the sum of cubes identity.

Example 3

easy
Expand (a+b)2(a+b)^2.

Example 4

easy
Expand (a−b)2(a-b)^2.

Example 5

easy
Factor a2−b2a^2 - b^2.

Example 6

easy
Compute (x+3)2(x+3)^2.

Example 7

easy
Factor x2−25x^2 - 25.

Example 8

easy
Compute (x−4)2(x-4)^2.

Example 9

easy
Use an identity to compute 99299^2.

Example 10

easy
Use an identity to compute 103×97103 \times 97.

Example 11

medium
Expand (2x+3)2(2x + 3)^2.

Example 12

medium
Factor 4x2−94x^2 - 9.

Example 13

medium
Expand (x+2)3(x+2)^3.

Example 14

medium
Factor x3−8x^3 - 8 (difference of cubes).

Example 15

medium
Simplify (a+b)2−(a−b)2(a+b)^2 - (a-b)^2.

Example 16

medium
Verify whether a2+b2=(a+b)2a^2 + b^2 = (a+b)^2 for a=1,b=2a=1, b=2.

Example 17

medium
Expand (x+y+z)2(x+y+z)^2.

Example 18

medium
Given a+b=5a+b=5 and ab=6ab=6, find a2+b2a^2+b^2.

Example 19

challenge
Prove a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2) by expansion.

Example 20

challenge
If x−1x=3x - \dfrac{1}{x} = 3, find x3−1x3x^3 - \dfrac{1}{x^3}.

Example 21

challenge
Show that (a2+b2)(c2+d2)=(ac+bd)2+(ad−bc)2(a^2+b^2)(c^2+d^2) = (ac+bd)^2 + (ad-bc)^2.

Example 22

medium
Factor x2+10x+25x^2 + 10x + 25 using a square identity.

Example 23

easy
Expand (x+7)2(x+7)^2.

Example 24

easy
Factor x2−49x^2 - 49.

Example 25

easy
Expand (x−6)2(x-6)^2.

Example 26

easy
Factor x2+12x+36x^2 + 12x + 36.

Example 27

easy
Compute (x+1)(x−1)(x+1)(x-1).

Example 28

medium
Expand (3x−4)2(3x - 4)^2.

Example 29

medium
Factor 25x2−1625x^2 - 16.

Example 30

medium
Factor x3+27x^3 + 27 as a sum of cubes.

Example 31

medium
Expand (x−3)3(x-3)^3.

Example 32

medium
If a+b=7a+b = 7 and ab=10ab = 10, find a2+b2a^2 + b^2.

Example 33

medium
Simplify (a+b)2+(a−b)2(a+b)^2 + (a-b)^2.

Example 34

medium
Factor 9−6x+x29 - 6x + x^2.

Example 35

medium
If x+1x=4x + \dfrac{1}{x} = 4, find x2+1x2x^2 + \dfrac{1}{x^2}.

Example 36

hard
Factor x6−y6x^6 - y^6 as completely as possible over the reals.

Example 37

hard
If a+b+c=0a + b + c = 0, show a3+b3+c3=3abca^3 + b^3 + c^3 = 3abc.

Example 38

hard
Expand (x+y+1)2(x + y + 1)^2.

Example 39

hard
Given a2+b2=13a^2 + b^2 = 13 and ab=6ab = 6, find a+ba + b (assume a,b>0a, b > 0).

Example 40

hard
If x+y=4x + y = 4 and xy=3xy = 3, find x3+y3x^3 + y^3.

Example 41

challenge
Prove (a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)(a+b+c)^3 = a^3 + b^3 + c^3 + 3(a+b)(b+c)(c+a).

Example 42

challenge
Show that n4+4n^4 + 4 factors over the integers for n≥1n \ge 1. Factor it.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

variable as generalizationidentity vs equationalgebraic pattern