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Lesson 4 of 8

Exploring Algebraic Identities · Lesson 4 of 8

More Identities

“More identities, because algebra apparently has several clever disguises.”

Learning Objectives

• Derive (a+b+c)². • Visualise the three-variable square identity. • Use the identity in calculation. • Understand a²−b²=(a+b)(a−b). • Use the difference-of-squares form for quick squaring.

The Square of Three Terms

Let d=b+c. Then (a+b+c)² becomes (a+d)², so we can use an identity we already know.

DerivationLaTeX
ExpandLaTeX

The identity has three square terms and three pairwise products, each doubled because each pair appears twice in full multiplication.

Geometric model of (a plus b plus c) squared A square of side a plus b plus c divided into nine regions: a squared, b squared, c squared, two ab rectangles, two bc rectangles and two ca rectangles. a² b² c² ab ab bc bc ca ca a b c a b c a + b + c (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Geometric model of (a+b+c)^2
Calculate 119²

Problem
Use 119=100+10+9.

  1. 1.119²=100²+10²+9²+2(100)(10)+2(10)(9)+2(9)(100).
  2. 2.=10000+100+81+2000+180+1800.
  3. 3.=14161.

Difference of Squares

IdentityLaTeX

Multiplying the right side gives a²−ab+ab−b², so the middle terms cancel.

A Useful Rearrangement

Rearranged identityLaTeX

The chapter connects this form with Śhrīdharāchārya and rapid calculation of squares. Choosing b as a small convenient difference can turn a square into an easy product.

Calculate 55²

Problem
Use the rearranged identity.

  1. 1.Choose a=55 and b=5.
  2. 2.55²=(60)(50)+25.
  3. 3.=3000+25=3025.
Area rearrangement for a squared equals (a plus b)(a minus b) plus b squared A square of side a is divided into two pieces and a smaller square of side b. The two pieces are rearranged into a rectangle with dimensions a plus b and a minus b, while the b squared square remains separate. Square of side a Rearranged areas a(a − b) b(a − b) b² a a a − b b rearrange a(a − b) b(a − b) a + b a − b (a + b)(a − b) + b² b b a² = (a + b)(a − b) + b²
Area rearrangement for a²=(a+b)(a-b)+b²

Practice Problems

Practice Problems
  1. Expand (p+3q+7r)².
  2. Expand (3x−2y+4z)².
  3. Find 117² using a suitable identity.
  4. Find 198² using a suitable identity.
  5. Find 1104² using a suitable identity.
  6. Use the difference-of-squares rearrangement to calculate 35², 65² and 85².
  7. Factor 16y²−24y+9.

Key Takeaways

Key Takeaways

• (a+b+c)² has three square terms and three doubled pair-products. • Substitution can reduce a new identity to a known one. • a²−b²=(a+b)(a−b). • Difference of squares is useful for factorisation and fast calculation. • Geometric models help explain identities, not merely memorise them.

Coming Next

Next, we use algebra tiles to see multiplication and factorisation as rectangular area.

Previous · Lesson 3

Factorisation of Algebraic Expressions Using Identities

Next · Lesson 5

Factorisation Using Algebra Tiles