Self-referential quadratic equation
I have written a quadratic equation with non-zero integer roots.
The sum of roots of this equation is equal to the answer to this question.
What is the product of roots of this same equation?
The answer is 4.
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2 solutions
@Shourya Pandey Very well! 👍
How do you infer that the product is A?
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Because the question asked is to find the product of roots of the equation.
how do you infer from ab = a + b that (a -1) + (b - 1) = 1
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Because 0 = a b − a − b = a ( b − 1 ) − b = a ( b − 1 ) − ( b − 1 ) − 1 = ( a − 1 ) ( b − 1 ) − 1 , so ( a − 1 ) ( b − 1 ) = 1 .
The answer will be the product of the two roots and the second line tells us that the sum of the roots is equal to the answer, meaning the sum is equal to the product and thus we can find out that it is either 0 or 2 on BOTH roots(the roots are the same). We are also given that it is non-zero, so we know that it is 2. From there, we can find out the answer - 2 times 2 is 4.
... and thus we can find out that it is either 0 or 2 on BOTH roots(the roots are the same).
How do you know that it's either 0 or 2 only? Why can't it be some other number?
Because the sum of the roots is equal to the product of the roots, we get that n^2 = 2n so n x n=2n and this can be solved as n(n-2)=0 which give 0 and 2.
This is assuming the roots are the same. For any roots that are DIFFERENT, it would not have a product equal to the sum.
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You didn't answer my question: How do you know it's either 0 or 2 only?
You didn't prove that it can't be any other number, but you just repeated the line of "Yeah, because sum = product"
The roots are the same. Let this root be n. Because the sum and product are equal, we have n^2 = 2n. This can be solved as
n^2 - 2n = 0 n(n-2) = 0 n = 0, 2
Thus the roots are either both 0 or both 2.
Let a and b be the integer roots to the equation. Then the answer A to this question is the sum of the roots, but it is also equal to the product of the roots. Therefore
a b = A = a + b
( a − 1 ) ( b − 1 ) = 1 , so
a − 1 = b − 1 = 1 or a − 1 = b − 1 = − 1 . So
( a , b ) = ( 2 , 2 ) , ( 0 , 0 ) , but the roots are non-zero. So a = b = 2 , and A = 4 .