GreenlightTestPrep wrote:
If \(x^{(3x^2-5x-7)}=x^{(2x^2-3x+1)}\), what is the sum of all possible values of \(x\)?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Note: This post is part of my Pro Tip series. You'll find my analysis and full solution below.
You may already be familiar with this question type (solving equations with variables in the exponents), but there's something a little different about this question.
With most other questions, the base is a constant (i.e., NOT a variable).
For example: \(5^{2x+1} = 5^{x-3}\)
Here, we can automatically conclude that \(2x+1 = x-3\), which we can then solve for \(x\)
This conclusion is based on the following property:
If \(b^x = b^y\), then \(x = y\) IMPORTANT: The above property has 3 provisos:
\(b \neq 0, b \neq 1\) and \(b \neq -1\) These provisos should make sense.
For example, \(1^x\) always equals 1. So, if an equation has \(1\) in the base, we really can't make any conclusions about the exponents.
So, if \(1^x = 1^3\), we can't then conclude that \(x=3\), since ANY value of \(x\) will satisfy the equation:
In other words, since \(1^x = 1^3=1^8=1^2=1^1=1^{-2} = 1...\) etc, we can't make any conclusions about the value of \(x\)
This same principle also applies to equations in which the base is \(0\) and \(-1\).
Given all of this, we need to consider a few different cases:
case i: \(x\) does
not equal \(0\), \(1\) or \(-1\)
This means we can safely apply the above
property to conclude that: \(3x^2-5x-7=2x^2-3x+1\)
Subtract \(2x^2\) from both sides of the equation to get: \(x^2-5x-7=-3x+1\)
Add \(3x\) to both sides of the equation to get: \(x^2-2x-7=1\)
Subtract \(1\) from both sides of the equation to get: \(x^2-2x-8=0\)
Factor: \((x-4)(x+2)=0\)
Solve: \(x=4\) and \(x=-2\)
case ii: \(x = 0\)
Since the base is one of the forbidden values (\(0\)), we can't apply the useful
property above.
Instead, we can plug \(x=0\) into the original equation to determine whether it's a solution.
When we do this we get: \(0^{3(0^2)-5(0)-7}=0^{2(0^2)-3(0)+1}\)
When we simplify, we get: \(0^{-7}=0^{1}\)
Evaluate to get: \(\frac{1}{0}=0\).
Since \(\frac{1}{0}\) is undefined,
\(x = 0\) is not a solution to the given equation. Aside: Since the question asks us to find the SUM of all possible solutions, we didn't really need to check whether \(x=0\) is a solution, since it won't have any effect on the SUM. I just wanted to demonstrate the steps necessary to determining ALL possible solutionscase iii: \(x = 1\)
Once again, to determine whether \(x=1\) is a possible solution, we'll plug \(x=1\) into the original equation.
We get: \(1^{3(1^2)-5(1)-7}=1^{2(1^2)-3(1)+1}\)
Simplify: \(1^{-9}=1^{0}\)
Evaluate to get: \(1=1\).
WORKS!So, \(x=1\) is another solution
Aside: We don’t need to actually test \(x = 1\), since 1 raised to any power will always be 1. I’m just showing all possible steps.case iv: \(x = -1\)
Once again, to determine whether \(x=-1\) is a possible solution, we'll plug \(x=-1\) into the original equation.
We get: \((-1)^{3(-1)^2)-5(-1)-7}=(-1)^{2(-1)^2)-3(-1)+1}\)
Simplify: \((-1)^{1}=(-1)^{6}\)
Evaluate to get: \(-1=1\).
DOESN'T WORKSo, \(x=-1\) is NOT a solution
The sum of all possible solutions = \(4 + (-2) + 1 = 3\)
Answer: C
Cheers,
Brent