2015 AMC 10A Problems/Problem 16
Problem
If , and
, what is the value of
?
Solution 1
Note that we can add the two equations to yield the equation
Moving terms gives the equation
We can also subtract the two equations to yield the equation
Moving terms gives the equation
Because we can divide both sides of the equation by
to yield the equation
Substituting this into the equation for that we derived earlier gives
Solution 2 (Algebraic)
Subtract from the LHS of both equations, and use difference of squares to yield the equations
and
.
It may save some time to find two solutions, and
, at this point. However,
in these solutions.
Substitute into
.
This gives the equation
which can be simplified to
.
Knowing and
are solutions now is helpful, as you divide both sides by
. This can also be done using polynomial division to find
as a factor. This gives
.
Because the two equations and
are symmetric, the
and
values are the roots of the equation, which are
and
.
Squaring these and adding the together gives
.
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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