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Re: What is the sum of all solutions to the equation [#permalink]
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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]
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]
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]
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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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Re: What is the sum of all solutions to the equation [#permalink]
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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:
Show: ::
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
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What is the sum of all solutions to the equation [#permalink]
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Why are we doing 0^(-6) = 1/(0^6) = 1/0, so 0^(-6) is UNDEFINED ?
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Re: What is the sum of all solutions to the equation [#permalink]
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Chaithraln2499 wrote:
Why are we doing 0^(-6) = 1/(0^6) = 1/0, so 0^(-6) is UNDEFINED ?


Sorry, I neglected to fix that certain part earlier.
I bet it in my response accordingly.

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Re: What is the sum of all solutions to the equation [#permalink]
Hi BrentGreenlightTestPrep
to clarify where is 0 in the final sum come from? from the factorisation? as testing of 0 is undefined so that doesn't count. Could you help clarify? Thanks Brent

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
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Re: What is the sum of all solutions to the equation [#permalink]
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Re: What is the sum of all solutions to the equation [#permalink]
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