SOLUTION: Use the quadratic formula to prove that the roots of ax²+bx+c=0 must have –b/a as their sum and c/a as their product. PLEASE HELP ME OUT!!!

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Use the quadratic formula to prove that the roots of ax²+bx+c=0 must have –b/a as their sum and c/a as their product. PLEASE HELP ME OUT!!!      Log On


   

Question 166121: Use the quadratic formula to prove that the roots of ax²+bx+c=0 must have –b/a as their sum and c/a as their product.
PLEASE HELP ME OUT!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember the quadratic formula is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 note: a%3C%3E0


Now let w=sqrt%28+b%5E2-4ac+%29 (to simplify things a bit)


So the quadratic formula becomes


x+=+%28-b+%2B-+w%29%2F%282a%29


which really breaks down to


x+=+%28-b+%2B+w%29%2F%282a%29 or x+=+%28-b+-+w%29%2F%282a%29


So the first root is x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29 and the second root is x%5B2%5D+=+%28-b+-+w%29%2F%282a%29


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Adding the Roots:



x%5B1%5D%2Bx%5B2%5D Now let's add the roots


%28-b+%2B+w%29%2F%282a%29%2B%28-b+-+w%29%2F%282a%29 Plug in x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29 and x%5B2%5D+=+%28-b+-+w%29%2F%282a%29


%28-b+%2B+w-b+-+w%29%2F%282a%29 Combine the fractions.


%28%28-b-b%29+%2B+%28w-w%29%29%2F%282a%29 Group like terms.


%28-2b%29%2F%282a%29 Combine like terms. Notice how the "w" terms cancel each other out completely.


-b%2Fa Reduce


So this shows us that the sum of the roots of any quadratic is -b%2Fa



Note: once you have the sum of the two roots you can find the vertex (even if you do NOT know the two roots all by themselves).


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Multiplying the Roots:



Remember, the first root is x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29 and the second root is x%5B2%5D+=+%28-b+-+w%29%2F%282a%29


x%5B1%5D%2Ax%5B2%5D Now multiply the roots.


%28%28-b+%2B+w%29%2F%282a%29%29%28%28-b+-+w%29%2F%282a%29%29 Plug in x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29 and x%5B2%5D+=+%28-b+-+w%29%2F%282a%29


%28%28-b+%2B+w%29%28-b+-+w%29%29%2F%28%282a%29%282a%29%29 Combine the fractions.


%28%28-b+%2B+w%29%28-b+-+w%29%29%2F%284a%5E2%29 Multiply %282a%29%282a%29 to get 4a%5E2


%28b%5E2%2Bbw-bw-w%5E2%29%2F%284a%5E2%29 FOIL the numerator


%28b%5E2-w%5E2%29%2F%284a%5E2%29 Combine like terms. Notice how the "bw" terms cancel out.


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Now remember all the way back to the start of the problem. We let w=sqrt%28+b%5E2-4ac+%29. So this means that

w%5E2=%28sqrt%28+b%5E2-4ac+%29%29%5E2=b%5E2-4ac


Or in other words: w%5E2=b%5E2-4ac

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%28b%5E2-%28b%5E2-4ac%29%29%2F%284a%5E2%29 Replace w%5E2 with b%5E2-4ac


%28b%5E2-b%5E2%2B4ac%29%2F%284a%5E2%29 Distribute the negative


%284ac%29%2F%284a%5E2%29 Combine like terms. The b%5E2 term is now gone.


c%2Fa Reduce


So this shows us that the product of two roots of any quadratic is c%2Fa