SOLUTION: Hello, 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 a

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Question 400987: Hello,
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.
I am not sure how to type the the index number, sorry. What I know is initially it looks like:
+%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 :)
Answer by jsmallt9(3759) About Me  (Show Source):
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.

Your expression:
%28x%2Bsqrt%283%29%29%28x-root%283%2C+3%29%29
is correct. But we need an equation, not an expression. The equation is simply
%28x%2Bsqrt%283%29%29%28x-root%283%2C+3%29%29+=+0
Now we just need it in ax%5E2%2Bbx%2Bc=0 form. For this we just multiply out the left side using FOIL:
x%5E2-x%2Aroot%283%2C+3%29%2B%28sqrt%283%29%29x-sqrt%283%29%2Aroot%283%2C+3%29=0
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.):

Factoring out x from the middle two terms we get:

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

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:
3%5E%281%2F2%29%2A3%5E%281%2F3%29
The rule for exponents when multiplying is to add the exponents. These exponents are fractions and to add fractions we need a common denominator:
3%5E%283%2F6%29%2A3%5E%282%2F6%29
Now we can multiply:
3%5E%28%283%2F6%29%2B%282%2F6%29%29
3%5E%285%2F6%29
We can now write this back in radical form:
root%286%2C+3%5E5%29
3%5E5+=+243 so this becomes:
root%286%2C+243%29
This makes our full, simplified equation:
x%5E2+%2B+%28-root%283%2C+3%29%2B+sqrt%283%29%29x+%2B+%28-root%286%2C+243%29%29=0
and ...
"a" being 1
"b" being %28-root%283%2C+3%29%2B+sqrt%283%29%29
and "c" being %28-root%286%2C+243%29%29

(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.)