MHB Sum and product of roots of quadratic equation

AI Thread Summary
The discussion focuses on deriving the sum and product of the roots of a quadratic equation using given relationships among the coefficients. The user initially finds the sum of the roots to be zero but becomes stuck in further calculations. Key equations are established, including relationships between the coefficients a, b, c, and d, leading to the conclusion that a + c equals 121. Ultimately, this results in the total sum a + b + c + d equating to 1210. The conversation emphasizes the importance of showing work to identify where confusion arises in solving quadratic equations.
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You've posted 4 questions, and I have given all 4 threads more useful titles, now I would ask that you show what you've tried in each thread. This way we can see where you're stuck.
 
Well i am stuck at this point
I found sum of zeroes in both cases and then got stuck.what should i do
 
Please post what you have so far. :)
 
We know:

$$c+d=10a$$

$$a+b=10c$$

Hence:

$$a+b+c+d=10(a+c)$$

We also know:

$$c^2-11b=a^2-11d$$

$$11d-11b=a^2-c^2$$

$$11(d-b)=(a+c)(a-c)$$

And we obtain by subtraction of the first 2 equations:

$$(d-b)-(a-c)=10(a-c)$$

Or:

$$d-b=11(a-c)$$

Hence:

$$121(a-c)=(a+c)(a-c)$$

Assuming \(a\ne c\) (otherwise \(d=b\) and the two quadratics are identical) there results:

$$a+c=121$$

Hence:

$$a+b+c+d=10\cdot121=1210$$
 

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