Which is bigger?

The idea

In general (a + b)² ≠ a² + b²; the term 2ab in the expansion of (a + b)² decides which of the two is larger.

A tempting shortcut

Many people think (a + b)² = a² + b². Is that right? Take a = 10 and b = 2.

Which is bigger, (10 + 2)² or 10² + 2²?

• (10 + 2)²
• 10² + 2²
• They are equal

Two bars

Compare the two bars for different a and b, even negative ones. Look at the gap between the bars: it is always 2ab.

The gap is 2ab

In general (a + b)² ≠ a² + b². The term 2ab decides which one is larger: (a + b)² = a² + b² + 2ab.

If 2ab is positive, (a + b)² is larger. If 2ab is negative, (a + b)² is smaller. If 2ab is zero, the two are equal.

Notes

(a + b)² ≠ a² + b² in general. The extra term 2ab decides which one is larger.

Check yourself

For a = 10 and b = 2, (a + b)² = 144 and a² + b² = 104. How much bigger is (a + b)²?

Answer: 40

144 − 104 = 40, and 2ab = 2 × 10 × 2 = 40. The gap is always 2ab.

For a = 3 and b = 4, which is larger?

For a = 5 and b = −2, which is larger?

When are (a + b)² and a² + b² equal?

• a² + b². 2ab = 24 is positive, so it makes (a + b)² larger, not smaller.
• (a + b)² — correct. 2ab = 24 is positive, so (a + b)² = 25 + 24 = 49 is larger than a² + b² = 25.
• They are equal. They differ by 2ab = 24.
• (a + b)². Look at 2ab = −20. It takes away from a² + b² = 29 and leaves 9.
• They are equal. They differ by 2ab = −20.
• a² + b² — correct. 2ab = 2 × 5 × (−2) = −20 is negative, so (a + b)² = 29 − 20 = 9 is smaller than a² + b² = 29.
• When 2ab = 0, that is when a = 0 or b = 0 — correct. Yes. The gap is 2ab, so the gap is zero exactly when a or b is zero.
• Always. Not always: for a = 10 and b = 2 they differ by 40.
• Never. Take a = 0: (0 + b)² = b² = 0² + b².