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 Dependent Variable

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Completing the Square

The essential part of completing the square is to recognize a perfect square trinomial when given its first two terms. For example, if we are given x2 + 6x, how do we recognize that these are the first two terms of the perfect square trinomial x2 + 6x + 9? To answer this question, recall that x2 + 6x + 9 is a perfect square trinomial because it is the square of the binomial x + 3:

(x + 3)2 = x2 + 2 Â· 3x + 32 = x2 + 6x + 9

Notice that the 6 comes frommultiplying 3 by 2 and the 9 comes from squaring the 3. So to find the missing 9 in x2 + 6x, divide 6 by 2 to get 3, then square 3 to get 9. This procedure can be used to find the last term in any perfect square trinomial in which the coefficient of x2 is 1.

Rule for Finding the Last Term

The last term of a perfect square trinomial is the square of one-half of the coefficient of the middle term. In symbols, the perfect square trinomial whose first two terms are

.

Review the rule for squaring a binomial: square the first term, find twice the product of the two terms, then square the last term. If you are still using FOIL to find the square of a binomial, it is time to learn the proper rule.

Example 1

Finding the last term

Find the perfect square trinomial whose first two terms are given.

Solution

a) One-half of 8 is 4, and 4 squared is 16. So the perfect square trinomial is x2 + 8x + 16.

b) One-half of -5 is , and squared is . So the perfect square trinomial is .

c) One-half of is , and squared is . So the perfect square trinomial is

d) One-half of is , and . So the perfect square trinomial is

Another essential step in completing the square is to write the perfect square trinomial as the square of a binomial. Recall that

a2 + 2ab + b2 = (a + b)2 and a2 - 2ab + b2 = (a - b)2.

Example 2

Factoring perfect square trinomials

Factor each trinomial.

Solution

a) The trinomial x2 + 12x + 36 is of the form a2 + 2ab + b2 with a = x and b = 6. So x2 + 12x + 36 = (x + 6)2. Check by squaring x + 6.

b) The trinomial is of the form a2 - 2ab + b2 with a = y and . So

Check by squaring .

c) The trinomial is of the form a2 - 2ab + b2 with a = z and . So