Square of a difference

The idea

(a – b)² = a² – 2ab + b², found by expanding with the distributive property or by writing a – b as a + (–b) in the identity for (a + b)².

A square inside a square

A square of side 60 has 60² = 3600 squares. A square of side 5 has 5² = 25. A square of side 55 sits in a corner of the big one, so two strips of 60 × 5 = 300 stick out.

How many squares are inside the 55 square?

• 3600 − 25
• 3600 − 300 − 300
• 3600 − 300 − 300 + 25

Take away the strips

Take away the right strip, then the bottom strip. Watch the corner. Then press the last button. Drag the purple dot to change b.

Add the corner back once

(a − b)² = a² − 2ab + b². Expand with the distributive property, or write a − b as a + (−b) in the identity for (a + b)².

55² = (60 − 5)² = 60² − (60 × 5) − (5 × 60) + 5² = 3600 − 300 − 300 + 25 = 3025

The corner 5 × 5 was taken away twice. We add it back once, so it is taken away only once.

(a − b)² = (a + (−b))² = a² + (−b)² + 2 × a × (−b) = a² + b² − 2ab NCERT calls this Identity 1B.

Notes

(a − b)² = a² − 2ab + b²: take away the two strips, then add the corner back once.

Check yourself

Use the identity to find 95². Write 95 as 100 − 5.

Answer: 9025

(100 − 5)² = 10000 − 1000 + 25 = 9025.

Which is the expansion of (x − 4)²?

Write (a − b)² as (a + (−b))² and use a² + 2ab + b². Which term changes sign?

Expand (3y − 2)². What is the number in front of y?

Answer: -12

(3y − 2)² = (3y)² − 2 × (3y × 2) + 2² = 9y² − 12y + 4.

• x² − 16. That leaves out the middle term, −2 × x × 4 = −8x. (x − 4)² is not x² − 4².
• x² − 8x − 16. The last term is (−4)² = +16. Minus times minus is plus: it is the corner added back.
• x² − 8x + 16 — correct. Yes! x² − 2 × x × 4 + 4² = x² − 8x + 16.
• a². The a stays a, so a² does not change.
• 2ab — correct. Yes! 2 × a × (−b) = −2ab. The other two terms stay the same.
• b². (−b)² = (−b) × (−b) = +b². Minus times minus is plus, so b² does not change.