SOLUTION: Without solving, find the product and the sum of the roots for 4x^2-7x+3 I know that a=4 b=-7 & c=3, I also have the equation, x^2+(-7)/4x +3/4 but I have no idea where to go fr

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Without solving, find the product and the sum of the roots for 4x^2-7x+3 I know that a=4 b=-7 & c=3, I also have the equation, x^2+(-7)/4x +3/4 but I have no idea where to go fr      Log On


   

Question 707166: Without solving, find the product and the sum of the roots for 4x^2-7x+3
I know that a=4 b=-7 & c=3, I also have the equation, x^2+(-7)/4x +3/4 but I have no idea where to go from there, So where do I go from there?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
You have the right values for a, b and c and that is all you need. You do not need to change the expression.

The sum of the roots of a quadratic is -b/a. So the sum of the roots for your expression will be -(-7)/(4) which simplifies to: 7/4

The product of the roots of a quadratic will be c/a. So the product of the roots to your expression will be (3)/(4) = 3/4

If you didn't know that the sum was -b/2a and/or that the product was c/a then it is not terribly difficult to figure this out on your own. From the Quadratic Formula we know that the roots of a general quadratic equation will be:
x%5B1%5D+=+%28-b+%2B+sqrt%28b%5E2-4ac%29%29%2F2a and x%5B2%5D+=+%28-b+-+sqrt%28b%5E2-4ac%29%29%2F2a
Splitting these into two separate fractions will help with the rest:
x%5B1%5D+=+%28-b%29%2F2a+%2B+sqrt%28b%5E2-4ac%29%29%2F2a and x%5B2%5D+=+%28-b%29%2F2a+-+sqrt%28b%5E2-4ac%29%2F2a

Now let's see what the sum looks like. If you look at x%5B1%5D and x%5B2%5D it should be easy to see that the second fractions will cancel each other out because one is positive and the other is negative. So when we add will will get:
%28-b%29%2F2a+%2B+%28%28-b%29%2F2a%29+=+%28-2b%29%2F2a+=+%28-b%29%2Fa

Now let's see what the product looks like:
. If we look at this we should be able to see that this fits the p%2Bq%29%28p-q%29 pattern. And from the pattern we know that the result is p%5E2-q%5E2. Using this pattern will save us time in multiplying:
%28%28-b%29%2F2a%29%5E2+-+%28sqrt%28b%5E2-4ac%29%29%2F2a%29%5E2%29
which simplifies as follows:
b%5E2%2F4a%5E2+-+%28b%5E2-4ac%29%2F4a%5E2%29
These have the same denominator so we can subtract them:
%28%28b%5E2%29+-+%28b%5E2-4ac%29%29%2F4a%5E2%29
Simplifying...
4ac%2F4a%5E2%29
c%2Fa
(Note: I have a terrible memory. I can never remember these sum and product formulas. I always go through the above to figure them out when I need them (like now)).