2020 AMC 10A Problems/Problem 14
Real numbers and
satisfy
and
. What is the value of
Solution 1
Continuing to combine
From the givens, it can be concluded that
. Also,
This means that
. Substituting this information into
, we have
. ~PCChess
Solution 2
As above, we need to calculate . Note that
are the roots of
and so
and
. Thus
where
and
as in the previous solution. Thus the answer is
.
Solution 3
Note that Now, we only need to find the values of
and
Recall that and that
We are able to solve the second equation, and doing so gets us
Plugging this into the first equation, we get
In order to find the value of we find a common denominator so that we can add them together. This gets us
Recalling that
and solving this equation, we get
Plugging this into the first equation, we get
Solving the original equation, we get ~emerald_block
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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