document.write( "Question 400987: Hello,\r
\n" ); document.write( "\n" ); document.write( "I need to write a quadratic equation in the variable x having the given numbers as solutions. Type the equation in standard form, ax^2 + bx + c = 0. The given numbers are two solutions: - sqrt 3, index 3 sqrt 3.
\n" ); document.write( "I am not sure how to type the the index number, sorry. What I know is initially it looks like:
\n" ); document.write( " $\"+%28x%2Bsqrt+3%29%2A%28x-index+3+sqrt+3%29+\"$ . I am not sure how to make this into a quadratic equation. Could you show me in complete detail how to do this please? Thank you :) \n" ); document.write( "

Algebra.Com's Answer #284006 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
I think by \"index 3 sqrt 3\" you mean $\"root%283%2C+3%29\"$ . If this is correct, then it is called the cube root of 3. It is not a square root of any kind.

\n" ); document.write( "Your expression:
\n" ); document.write( " $\"%28x%2Bsqrt%283%29%29%28x-root%283%2C+3%29%29\"$
\n" ); document.write( "is correct. But we need an equation, not an expression. The equation is simply
\n" ); document.write( " $\"%28x%2Bsqrt%283%29%29%28x-root%283%2C+3%29%29+=+0\"$
\n" ); document.write( "Now we just need it in $\"ax%5E2%2Bbx%2Bc=0\"$ form. For this we just multiply out the left side using FOIL:
\n" ); document.write( " $\"x%5E2-x%2Aroot%283%2C+3%29%2B%28sqrt%283%29%29x-sqrt%283%29%2Aroot%283%2C+3%29=0\"$
\n" ); document.write( "What follows will make more sense if we rewrite the equation as additions. (Also, the $\"ax%5E2%2Bbx%2Bc=0\"$ form is written as additions.):
\n" ); document.write( "

\n" ); document.write( "Factoring out x from the middle two terms we get:
\n" ); document.write( "

\n" ); document.write( "If you have not yet learned about fractional exponents then the equation above is your answer with...
\n" ); document.write( "\"a\" being 1
\n" ); document.write( "\"b\" being $\"%28-root%283%2C+3%29%2B+sqrt%283%29%29\"$
\n" ); document.write( "and \"c\" being $\"%28-sqrt%283%29%2Aroot%283%2C+3%29%29\"$

\n" ); document.write( "If you do know about fractional exponents then we can simplify $\"%28sqrt%283%29%2Aroot%283%2C+3%29%29\"$ . Rewriting this with fractional exponents we get:
\n" ); document.write( " $\"3%5E%281%2F2%29%2A3%5E%281%2F3%29\"$
\n" ); document.write( "The rule for exponents when multiplying is to add the exponents. These exponents are fractions and to add fractions we need a common denominator:
\n" ); document.write( " $\"3%5E%283%2F6%29%2A3%5E%282%2F6%29\"$
\n" ); document.write( "Now we can multiply:
\n" ); document.write( " $\"3%5E%28%283%2F6%29%2B%282%2F6%29%29\"$
\n" ); document.write( " $\"3%5E%285%2F6%29\"$
\n" ); document.write( "We can now write this back in radical form:
\n" ); document.write( " $\"root%286%2C+3%5E5%29\"$
\n" ); document.write( " $\"3%5E5+=+243\"$ so this becomes:
\n" ); document.write( " $\"root%286%2C+243%29\"$
\n" ); document.write( "This makes our full, simplified equation:
\n" ); document.write( " $\"x%5E2+%2B+%28-root%283%2C+3%29%2B+sqrt%283%29%29x+%2B+%28-root%286%2C+243%29%29=0\"$
\n" ); document.write( "and ...
\n" ); document.write( "\"a\" being 1
\n" ); document.write( "\"b\" being $\"%28-root%283%2C+3%29%2B+sqrt%283%29%29\"$
\n" ); document.write( "and \"c\" being $\"%28-root%286%2C+243%29%29\"$

\n" ); document.write( "(Note: The \"b\", $\"%28-root%283%2C+3%29%2B+sqrt%283%29%29\"$ , cannot be simplified, even if you use fractional exponents. They are not like terms and cannot be transformed into like terms. So we can never add them.) \n" ); document.write( "

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