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Squaring a SumTo find (a + b)^{2}, the square of a sum, we can use FOIL on (a + b)(a + b):
You can use the result a^{2} + 2ab + b^{2} that we obtained from FOIL to quickly find the square of any sum. To square a sum, we square the first term (a^{2}), add twice the product of the two terms (2ab), then add the square of the last term (b^{2}).
Rule for the Square of a Sum (a + b)^{2} = a^{2} + 2ab + b^{2} In general, the square of a sum (a + b)^{2 }is not equal to the sum of the squares a^{2} + b^{2}. 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) (2y^{4} + 3)^{2} Solution
b) (2w + 3)^{2} = (2w)^{2} + 2(2w)(3) + 3^{2} = 4w^{2} + 12w + 9 c) (2y^{4} + 3)^{2} = (2y^{4})^{2} + 2(2y^{4})(3) + 3^{2} = 4y^{8} + 12y^{4} + 9 Caution Squaring x + 5 correctly, as in Example 2(a), gives us the identity (x + 5)^{2} = x^{2} + 10x + 25, which is satisfied by any x. If you forget the middle term and write (x + 5)^{2} = x^{2} + 25, then you have an equation that is satisfied only if x = 0. Helpful hint You can use
to learn a trick for squaring a number that ends in 5. For example, to find 25^{2}, find 20 Â· 30 + 25 or 625. More simply, to find 35^{2}, find 3 Â· 4 = 12 and follow that by 25: 35^{2} = 1225. |