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

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Question 166121: Use the quadratic formula to prove that the roots of ax�+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) (Show Source):
You can put this solution on YOUR website!
Remember the quadratic formula is:

note:

Now let (to simplify things a bit)

So the quadratic formula becomes

which really breaks down to

or

So the first root is and the second root is

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Adding the Roots:

Now let's add the roots

Plug in and

Combine the fractions.

Group like terms.

Combine like terms. Notice how the "w" terms cancel each other out completely.

Reduce

So this shows us that the sum of the roots of any quadratic is

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 and the second root is

Now multiply the roots.

Plug in and

Combine the fractions.

Multiply to get

FOIL the numerator

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 . So this means that

Or in other words:

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Replace with

Distribute the negative

Combine like terms. The term is now gone.

Reduce

So this shows us that the product of two roots of any quadratic is