document.write( "Question 890611: If i have roots or solution m and n, what can i do to identify the standard form of the given roots?
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Algebra.Com's Answer #539105 by KMST(5328) $\"\"$ $\"About$

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IN GENERAL:
\n" ); document.write( "A quadratic equation is an equation of degree $\"2\"$ , so you know that there will be an $\"x%5E2\"$ somewhere.
\n" ); document.write( "If you know that $\"m\"$ and $\"n\"$ are solutions (or roots) of a quadratic equation,
\n" ); document.write( "you can write a quadratic equation with those solutions as
\n" ); document.write( " $\"%28x-m%29%28x-n%29=0\"$ .
\n" ); document.write( "That is the equation in factored form.
\n" ); document.write( "If you multiply the factors $\"%28x-m%29\"$ and $\"%28x-n%29\"$ and simplify the result,
\n" ); document.write( "you get a more elegant quadratic equation that your teacher will like better.
\n" ); document.write( "There are infinite quadratic equations with those roots (or solutions).
\n" ); document.write( "You can transform a quadratic equation into an equivalent one
\n" ); document.write( "(one with exactly the same solutions, no more or less solutions, no different solutions)
\n" ); document.write( "by multiplying both sides of the equation by a number other than zero:
\n" ); document.write( "Take any $\"a%3C%3E0\"$ , and
\n" ); document.write( " $\"%28x-m%29%28x-n%29=0\"$ ---> $\"a%2A%28x-m%29%28x-n%29=a%2A0\"$ ---> $\"a%28x-m%29%28x-n%29=0\"$
\n" ); document.write( "(If you multiplied both sides times zero you would end up with $\"0=0\"$ that is not much of an equation).
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\n" ); document.write( "EXAMPLE:
\n" ); document.write( "For example, if the solutions or roots of a quadratic equation are $\"-3\"$ and $\"1%2F2\"$ , I would write
\n" ); document.write( " $\"%28x-%28-3%29%29%28x-1%2F2%29=0\"$ ---> $\"%28x%2B3%29%28x-1%2F2%29=0\"$ ---> $\"x%5E2-%281%2F2%29x%2B3x-3%2F2=0\"$ ---> $\"x%5E2%2B%285%2F2%29x-3%2F2=0\"$ ---> $\"2%2A%28x%5E2%2B%285%2F2%29x-3%2F2%29=2%2A0\"$ ---> $\"highlight%282x%5E2%2B5x-3=0%29\"$
\n" ); document.write( "All those equations are equivalent, but the last one is the simplest and most elegant of them.
\n" ); document.write( "I could multiply both sides times a non-zero number, let's say $\"100\"$ , to get an elegant but unnecessarily complicated equation:
\n" ); document.write( " $\"200x%5E2%2B500x-300=0\"$ .
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\n" ); document.write( "EXTRA:
\n" ); document.write( "Also, with the same roots I can write equations of a degree higher than $\"2\"$ (not quadratic),
\n" ); document.write( "by multiplying both sides times some polynomial, like $\"x\"$ or $\"%28x%5E2%2B1%29\"$ . \n" ); document.write( "

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