2023 AMC 12A Problems/Problem 19
Contents
Problem
What is the product of all solutions to the equation
Solution 1
For , transform it into . Replace with . Because we want to find the product of all solutions of , it is equivalent to finding the exponential of the sum of all solutions of . Change the equation to standard quadratic equation form, the term with 1 power of is canceled. By using Vieta, we see that since there does not exist a term, and .
~plasta
Solution 2 (Same idea as Solution 1 with easily understand steps)
Rearranging it give us:
let be , we get
by Vieta's Formulas,
~lptoggled
Solution 3
Similar to solution 1, change the bases first Cancel and cross multiply to get Simplify to get The sum of all possible is 0, thus the product of all solutions of is
~dwarf_marshmallow
Solution 4
We take the reciprocal of both sides: Using logarithm properties, we have Simplify to obtain from which we have
~MLiang2018
Solution 5(Similar to solution 4)
First, we take the reciprocal of both sides. We get Flip the logarithms to get
Now we can use . We get The and terms cancel, giving so now we are sure that , so the only solution is .
~Yrock
Video Solution 1 by OmegaLearn
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 12 Problems and Solutions |
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