Difference between revisions of "1983 AIME Problems/Problem 3"
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=== Solution 2 === | === Solution 2 === | ||
− | We begin by noticing that the polynomial on the left is <math>15</math> less than the polynomial under the radical sign. Thus: <cmath>(x^2+ | + | We begin by noticing that the polynomial on the left is <math>15</math> less than the polynomial under the radical sign. Thus: <cmath>(x^2+ 18x + 45) - 2\sqrt{x^2+18x+45} - 15 = 0.</cmath> Letting <math>n = \sqrt{x^2+18x+45}</math>, we have <math>n^2-2n-15 = 0 \Longrightarrow (n-5)(n+3) = 0</math>. Because the square root of a real number can't be negative, the only possible <math>n</math> is <math>5</math>. |
− | Substituting that in, we have <cmath>\sqrt{x^2+ | + | Substituting that in, we have <cmath>\sqrt{x^2+18x+45} = 5 \Longrightarrow x^2 + 18x + 45 = 25 \Longrightarrow x^2+18x+20=0.</cmath> |
And by [[Vieta's formulas]], the product of the roots is <math>\boxed{020}</math>. | And by [[Vieta's formulas]], the product of the roots is <math>\boxed{020}</math>. |
Revision as of 20:17, 30 January 2017
Problem
What is the product of the real roots of the equation ?
Solutions
Solution 1
If we expand by squaring, we get a quartic polynomial, which isn't always the easiest thing to deal with.
Instead, we substitute for
and our equation becomes
.
Now we can square; solving for , we get
or
(The second solution is extraneous since
is positive (plugging in
as
, we get
, which is obviously not true)).So, we have
as the only solution for
. Substituting
back in for
,
![$x^2+18x+30=10 \Longrightarrow x^2+18x+20=0.$](http://latex.artofproblemsolving.com/4/f/b/4fbfb8c3979ed14768e003b2256da6a1b2b0fd3e.png)
By Vieta's formulas, the product of the roots is .
Solution 2
We begin by noticing that the polynomial on the left is less than the polynomial under the radical sign. Thus:
Letting
, we have
. Because the square root of a real number can't be negative, the only possible
is
.
Substituting that in, we have
And by Vieta's formulas, the product of the roots is .
See Also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |