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What is the sum of all solutions to the equation

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What is the sum of all solutions to the equation [#permalink] New post 02 Jan 2017, 12:43
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What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

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Answer:
[Reveal] Spoiler:
5

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Re: What is the sum of all solutions to the equation [#permalink] New post 03 Jan 2017, 10:22
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GreenlightTestPrep wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

* Kudos for all correct solutions


Answer:
[Reveal] Spoiler:
5


IMPORTANT: If b^x = b^y, then x = y, as long as b ≠ 0, b ≠ 1 and b ≠ -1
For example, if we have 1^x = 1^y, we cannot conclude that x = y, since 1^x equals 1^y FOR ALL values of x and y. For example, 1² = 1³, but we can't conclude that 2 = 3.

So, let's first see what happens when the base (x) equals 0, 1 and -1

If x = 0, then we have: 0^(2(0²) + 4(0) – 6) = 0^(0² + 8(0) + 6)
Simplify: 0^(-6) = 0^6
Evaluate: 0 = 0
So, x = 0 is one solution to the equation (yes, I know that x = 0 does not change the SUM of the solutions. I just want to show all of the possible considerations)

If x = 1, then we have: 1^(2(1²) + 4(1) – 6) = 1^(1² + 8(1) + 6)
Simplify: 1^0 = 1^15
Evaluate: 1 = 1
So, x = 1 is another solution to the equation

If x = -1, then we have: (-1)^[2(-1)² + 4(-1) – 6] = (-1)^[(-1)² + 8(-1) + 6]
Simplify: (-1)^(-8) = (-1)^(-1)
Evaluate: 1 = -1
So, x = -1 is NOT a solution to the equation

Now let's assume that x ≠ 0, x ≠ 1 and x ≠ -1 and look for other x-values that satisfy the given equation.
Given: x^(2x² + 4x – 6) = x^(x² + 8x + 6)
Since the bases are the same, we can write: 2x² + 4x – 6 = x² + 8x + 6
Rearrange to get: x² - 4x – 12 = 0
Factor to get: (x - 6)(x + 2) = 0
So, x = 6 and x = -2 are also solutions to the equation.

So, the solutions are x = 0, x = 1, x = 6, and x = -2
0 + 1 + 6 + (-2) = 5

Answer: 5
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Re: What is the sum of all solutions to the equation [#permalink] New post 17 Aug 2018, 01:35
isn't it contradictory since you mentioned that the base cannot be 0,1,or-1 yet you tested X for all these values. Appreciate your clarification. Thanks
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Re: What is the sum of all solutions to the equation [#permalink] New post 17 Aug 2018, 06:45
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Runnyboy44 wrote:
isn't it contradictory since you mentioned that the base cannot be 0,1,or-1 yet you tested X for all these values. Appreciate your clarification. Thanks


Be careful; I didn't say that the base cannot be 0, 1 or -1.
I said that, if the base equals 0, 1 or -1, then the rule does not necessarily apply.

For example, let's say we're told that b^x = b^y
Can we conclude that x = y?
No. We can only conclude that x = y IF we are certain that b does not equal 0, 1 or -1.

So, before we can make any conclusions about the exponents being equal, we must first ensure that the base does not equal 0, 1 or -1

Does that help?

Cheers,
Brent
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Re: What is the sum of all solutions to the equation [#permalink] New post 17 Aug 2018, 16:31
Equate the x below to get x=1 and equate the equations above to get the solutions -6, 2.
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Re: What is the sum of all solutions to the equation [#permalink] New post 19 Oct 2018, 10:15
I did not understand how x= 1 and 0.. Though I got the other two values 6 and -2. Plz explain
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Re: What is the sum of all solutions to the equation [#permalink] New post 19 Oct 2018, 23:07
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Reetika1990 wrote:
I did not understand how x= 1 and 0.. Though I got the other two values 6 and -2. Plz explain


whatever be the exponent, if the base is 0 or 1, answer will always be 0 or 1 respectively except when power is negative..
for example in this equation..

\(x^{2x² + 4x – 6} = x^{x² + 8x +6}\)
x=0...
\(0^{-6}=0^6....undefined=0\), so 0 may not be a value
x=1
\(1^{2*1^2+4*1-6}=1^{1^2+8*1+6}.......1^0=1^{15}....1=1\)...yes
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: What is the sum of all solutions to the equation [#permalink] New post 19 Oct 2018, 23:21
Expert's post
GreenlightTestPrep wrote:
GreenlightTestPrep wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

* Kudos for all correct solutions


Answer:
[Reveal] Spoiler:
5



So, let's first see what happens when the base (x) equals 0, 1 and -1

If x = 0, then we have: 0^(2(0²) + 4(0) – 6) = 0^(0² + 8(0) + 6)
Simplify: 0^(-6) = 0^6
Evaluate: 0 = 0
So, x = 0 is one solution to the equation (yes, I know that x = 0 does not change the SUM of the solutions. I just want to show all of the possible considerations)




Hi @GreenlightTestPrep,

excellent question..
I do agree 0 will not make a difference to the solution but 0 may not be a value of x here because one side becomes 0 to the power of -6, a negative number..
\(0^{-6}=\frac{1}{0^6}=\frac{1}{0}\), an undefined value
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: What is the sum of all solutions to the equation   [#permalink] 19 Oct 2018, 23:21
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