document.write( "Question 157297: 2. In order to fully understand what I'm doing here, I need to have it explained to me how to factor a polynomial of the form ax^2 + bx + c when a is not equal to 1. I know there is the grouping approach as well as reversing FOIL and I want to be able to see an example. Which approach would be the best? Could somebody explain this to me as well? Thank you so very much, I truly need guidance on this. Thank you. \n" ); document.write( "
Algebra.Com's Answer #115948 by gonzo(654)\"\" \"About 
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to create an example, you can work backwards.
\n" ); document.write( "take (7x+5) * (5x-3) for example.
\n" ); document.write( "the quadratic equation would then be -3*5 -21x + 25x + 35x^2
\n" ); document.write( "which would then equal 35x^2 + 25x - 21x - 15
\n" ); document.write( "which would then equal 35x^2 + 4x - 15
\n" ); document.write( "how would you solve this equation?
\n" ); document.write( "if you couldn't figure out the combinations that would work fairly quickly, then the best approach would be to use the quadratic equation of \"%28-b+%2B+%28b%5E2-4ac%29%5E%281%2F2%29%29+%2F+2a\" and \"%28-b+-+%28b%5E2-4ac%29%5E%281%2F2%29%29+%2F+2a\"
\n" ); document.write( "looking at the equation \"35x%5E2%2B4x-15\" yields the following conclusions.
\n" ); document.write( "it's been reduced to its simplest form.
\n" ); document.write( "there are no common factors that can be eliminated.
\n" ); document.write( "the c factors would be a combination of (1 and 15) or (3 and 5)
\n" ); document.write( "the a factors would be a combination of (35 and 1) or (5 and 7)
\n" ); document.write( "try (35 and 1) for the a factors first combined with (1 and 15) for the c factors.
\n" ); document.write( "cross multiplying the a factors with the c factors yields the following taking into account that one of the c factors has to be minus:
\n" ); document.write( "35x * 1 = -35x or 35x
\n" ); document.write( "35x * 15 = 525x or -525x
\n" ); document.write( "possibilities are 35x, -35x, 525x, -525x.
\n" ); document.write( "none of these can be added together to get a b factor of 4.
\n" ); document.write( "conclusion is (35 and 1) combined with (1 and 15) won't work.
\n" ); document.write( "intuition might tell you to try (5 and 7) for the a factors and (3 and 5) for the c factors next. this will also shorten the time required to explain this.
\n" ); document.write( "remember one of the c factors has to be minus, either 3 or 5.
\n" ); document.write( "5x * 3 15x or -15x
\n" ); document.write( "5x * 5 = 25x or -25x
\n" ); document.write( "7x * 3 = 21x or -21x
\n" ); document.write( "7x * 5 = 35x or -35x
\n" ); document.write( "possibilities are 15x, -15x, 25x, -25x, 21x, -21x, 35x, -35x.
\n" ); document.write( "out of all these, the combination that will yield a b factor of 4 is 25x - 21x.
\n" ); document.write( "you answer has to be some combination of (5x * 5) and (7x * 3)
\n" ); document.write( "in order to get that cross multiplication you would need to have factors of either (5x-3) or (5x+3) and (7x-5) or (7x+5).
\n" ); document.write( "intuition will tell you that the 25 must be positive to get a positive b factors, so you would choose (7x+5) * (5x-3) because your cross multiplication then gets you 25x - 21x.
\n" ); document.write( "cross multiplying those factors will get you -15 -21x + 25x + 35x^2.
\n" ); document.write( "combining and moving terms gets you 35x^2 +4x -15 which is the equation you want.\r
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