Spot the perfect square

The idea

An expression that can be written in the form a² + 2ab + b² factorises as (a + b)²; a factor common to all its terms may be taken out first.

A familiar shape

Look at x² + 4x + 4. The first term is x² and the last term is 4 = 2². Does the middle term 4x remind you of 2ab?

x² + 4x + 4 = (x + ?)². What goes in the brackets?

• 2
• 4
• 1

Find a and b

Match each expression with a² + 2ab + b². Pick a and b, and let the three checks tell you if you are right.

Read it backwards

An expression of the form a² + 2ab + b² factorises as (a + b)². If all terms have a common factor, take it out first.

x² + 4x + 4 = x² + 2(x)⁠(2) + 2² = (x + 2)² So (x + 2) is a factor of x² + 4x + 4. 50p² + 60pq + 18q² = 2(25p² + 30pq + 9q²) = 2(5p + 3q)²

Notes

If an expression looks like a² + 2ab + b², it is (a + b)². Take out a common factor first when the squares are not obvious.

Check yourself

x² + 6x + 9 = ?

x² + 10x + 25 = (x + ?)². Which number?

Answer: 5

b² = 25 gives b = 5, and 2ab = 2 × x × 5 = 10x. So the answer is (x + 5)².

Why is x² + 5x + 4 not a perfect square?

Take out the common factor first: 2x² + 12x + 18 = ?

• (x + 3)² — correct. Yes: x² + 2(x)⁠(3) + 3² = (x + 3)².
• (x + 6)². (x + 6)² = x² + 12x + 36.
• (x + 9)². (x + 9)² = x² + 18x + 81.
• Because it has three terms. a² + 2ab + b² also has three terms.
• b² = 4 gives b = 2, but then 2ab = 4x, not 5x — correct. Yes. All three checks must pass. Here the check for 2ab fails.
• Because the middle term has an odd number. What matters is whether the middle term equals 2ab, not whether its number is odd.
• (2x + 3)². (2x + 3)² = 4x² + 12x + 9.
• 2(x + 9)². (x + 9)² = x² + 18x + 81, so 2(x + 9)² is much too big.
• 2(x + 3)² — correct. Yes: 2x² + 12x + 18 = 2(x² + 6x + 9), and x² + 6x + 9 = (x + 3)². So it is 2(x + 3)².