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A \"pure\" quadratic has the form:
\n" ); document.write( "ax^2 + bx + c
\n" ); document.write( "To put this in broad, general (and crude) terms, a quadratic is:
\n" ); document.write( "\"Some number, 'a', times a perfect square plus some number. 'b', times whatever-is being squared-next to-the-a (x) plus some other number, 'c'\"
\n" ); document.write( "So can we make 9x^4+20x^2+12 fit this pattern? Let's look at this expression term-by-term:
\n" ); document.write( "9x^4
\n" ); document.write( "Can this be expressed as \"Some number, 'a', times a perfect square\"? Answer: Yes, in two ways:
\n" ); document.write( "9x^4 = 9*(x^2)^2
\n" ); document.write( "or
\n" ); document.write( "1*(3x^2)^2
\n" ); document.write( "Next
\n" ); document.write( "10x^2
\n" ); document.write( "Can this be expressed as \"some number. 'b', times whatever-is being squared-next to-the-a\"? The answer depends on which of the expressions we use for 9x^4.
\n" ); document.write( "If we use 9*(x^2)^2 then \"whatever-is being squared-next to-the-a\" would be x^2. Can we express 20x^2 as some number times x^2? Obviously yes: 20*x^2.
\n" ); document.write( "If we try to use 1*(3x^2)^2 for the 9x^4 then \"whatever-is being squared-next to-the-a\" would be 3x^2. Can we express 20x^2 as some number times 3x^2? Although not very obvious, the answer is yes: 20x^2 = (20/3)*3x^2
\n" ); document.write( "And of course 12 can be our 'c'. So there are two ways to express 9x^4+20x^2+12 in quadratic form:
\n" ); document.write( "9x^4+20x^2+12 = 9*(x^2)^2 + 20*x^2 + 12 with the a = 9, b = 20 and the c = 12
\n" ); document.write( "or
\n" ); document.write( "9x^4+20x^2+12 = 1*(3x^2)^2 + (20/3)*3x^2 + 12 with the a = 1, b = 20/3 and the c = 12.
\n" ); document.write( "If you really want these to look like a quadratic, then you can use a temporary variable. Make the temporary variable equal to \"whatever-is being squared-next to-the-a\". So for 9*(x^2)^2 + 20*x^2 + 12, let q = x^2. This makes the expression: 9q^2 + 20q + 12. For 1*(3x^2)^2 + (20/3)*3x^2 + 12, let q = 3x^2. This makes the expression: q^2 + (20/3)q + 12. With the temporary variable the expression might be easier to factor or the Quadratic Formula might be easier to use. \n" ); document.write( "