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Squaring a Sum

To find (a + b)², the square of a sum, we can use FOIL on (a + b)(a + b):

(a + b)(a + b)	= a² + ab + ab + b²
	= a² + 2ab + b²

You can use the result a² + 2ab + b² that we obtained from FOIL to quickly find the square of any sum. To square a sum, we square the first term (a²), add twice the product of the two terms (2ab), then add the square of the last term (b²).

Rule for the Square of a Sum

(a + b)² = a² + 2ab + b²

In general, the square of a sum (a + b)²is not equal to the sum of the squares a² + b². The square of a sum has the middle term 2ab.

Helpful hint

To visualize the square of a sum, draw a square with sides of length a + b as shown.

The area of the large square is (a + b)². It comes from four terms as stated in the rule for the square of a sum.

Example 2

Squaring a sum

Square each sum, using the new rule.

a) (x + 5)²

b) (2w + 3)²

c) (2y⁴ + 3)²

Solution

a) (x + 5)²	= x² +	2(x)(5)	+ 5²	= x² + 10x + 25
	↑ Square of first	↑ Twice the product	↑ Square of last

b) (2w + 3)² = (2w)² + 2(2w)(3) + 3² = 4w² + 12w + 9

c) (2y⁴ + 3)² = (2y⁴)² + 2(2y⁴)(3) + 3² = 4y⁸ + 12y⁴ + 9

Caution

Squaring x + 5 correctly, as in Example 2(a), gives us the identity (x + 5)² = x² + 10x + 25, which is satisfied by any x. If you forget the middle term and write (x + 5)² = x² + 25, then you have an equation that is satisfied only if x = 0.

Helpful hint

You can use

(x + 5)²	= x² + 10x + 25
	= x(x + 10) + 25

to learn a trick for squaring a number that ends in 5. For example, to find 25², find 20 Â· 30 + 25 or 625. More simply, to find 35², find 3 Â· 4 = 12 and follow that by 25: 35² = 1225.