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4x2 + 8x + 16
\n" ); document.write( " x2 +9x + 9
\n" ); document.write( " x2 -6x +36
\n" ); document.write( "4x2 - 12x + 9
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\n" ); document.write( "In order for these trinomials to be perfect squares, their first terms and their last terms
\n" ); document.write( "need to be perfect squares and positive. That's not a help because 4x^2, 16, x^2, 9, and 36
\n" ); document.write( "are all perfect squares and all are positive. That means all the trinomials could be perfect squares.
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\n" ); document.write( "Let's look at the second and third trinomials. In the second, the + 9 factors into either
\n" ); document.write( "+ 3 times +3 or -3 times -3. But there is no way that either +3x and +3x or -3x and -3x can
\n" ); document.write( "be added to give the middle term +9x of the trinomial (+3x and -3x are the products of
\n" ); document.write( "the square root of the last term in the trinomial times the square root of the first term
\n" ); document.write( "of the trinomial.) Because you can't combine the possible products of the square root of the
\n" ); document.write( "last term and the first term to get the middle term, that eliminates the the trinomial from
\n" ); document.write( "being a perfect square.
\n" ); document.write( ".
\n" ); document.write( "The same thing can be said to happen to the third trinomial. The 36 factors into +6 times +6
\n" ); document.write( "or -6 times -6. But there is no way that +6x and +6x can be combined to give the middle term
\n" ); document.write( "of -6x. The same can be said of -6x and -6x. They can't be combined to give the middle
\n" ); document.write( "term of -6x. Since neither will work, this eliminates the third equation from being a perfect
\n" ); document.write( "square.
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\n" ); document.write( "Now, let's do the first trinomial. Notice that the square root of the first term is
\n" ); document.write( "2x. Also note that the square root of +16 is either +4 or -4. Now form the product of
\n" ); document.write( "the square roots of these two terms. First the product of 2x and +4 is +8x. Is there any
\n" ); document.write( "way that +8x and + 8x can be combined to give the middle term of the trinomial +8x? No.
\n" ); document.write( "So how about doing the same thing with the -4 times 2x to get -8x? You can't get -8x and
\n" ); document.write( "-8x to add up to +8x either.
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\n" ); document.write( "So, finally let's try the fourth trinomial of 4x2 - 12x + 9. Again, the square root of the
\n" ); document.write( "first term is 2x and the square root of the last term is either +3 or -3. Multiply the
\n" ); document.write( "+3 times the 2x and get +6x. Is there anyway that +6x and +6x can be combined to give the
\n" ); document.write( "trinomial's middle term of -12x? Nope, but how about the product of -3 and 2x? That product
\n" ); document.write( "is -6x. That is -6x. And when you add -6x and -6x you get the middle term of the trinomial.
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\n" ); document.write( "The factors of the fourth trinomial are both (2x - 3) and (2x - 3). When you multiply
\n" ); document.write( "these two together you get back to the original trinomial. Therefore, the answer to
\n" ); document.write( "your question is the last of the four trinomials that you were given as potential
\n" ); document.write( "answers.
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\n" ); document.write( "Hope this helps you to see how to examine each trinomial to see if it is a perfect
\n" ); document.write( "square. \n" ); document.write( "