JEE Mains 2016 Mathematics 36

Algebra Level 4

$\large (x^2-5x+5)^{x^2+4x-60} = 1$

Find the sum of all values of $x$ satisfying the equation above.

6 3 5 -4

1 solution

Shivam Mishra
Apr 10, 2016

Well, this could be solved by considering the following cases:

(i) when the expression becomes $1^{n}$
(ii)when the expression becomes $n^0$ provided ' $n$ ' is not equal to $0$
(iii)when the expression becomes $(-1)^{even}$

After considering the above cases we obtain the following solutions $(4,1)$ , $(6,{-10})$ , $(2)$ .

In the last solution pair $3$ is excluded as the exponent to $-1$ turns out to be odd negative.

Now adding the solutions gives us the answer as $\boxed{3}$ .

Moderator note:

Great. Using the classification of solutions to $a^b = 1$ , we can figure out the solutions in a sensible manner.

This is one of the most common variant of problems after $i^i$ .Anyways, good job (+1) :)

Rohit Udaiwal - 5 years, 2 months ago

Thanks and yes these questions are pretty common.

shivam mishra - 5 years, 2 months ago

How to find (-1)^even ?

Daniel Sugihantoro - 5 years, 2 months ago

@Daniel Sugihantoro to find that you need to equate $x^2-5x+5$ to ${-1}$ and solve the quadratic thus obtained.Now plug in the solutions obtained in the quadratic in the exponent.One of the values obtained would be odd and negative.This would be rejected as the it would turn the whole expression into ${-1}$ which is not what we want.Thus,only ${2}$ would be included in our solution.

shivam mishra - 5 years, 2 months ago

Oh i understand now, thank you

Daniel Sugihantoro - 5 years, 2 months ago

FYI To start a new line, just leave 3 empty spaces at the end of the sentence. I've edited your (i), (ii), (iii) points to display nicer.

Calvin Lin Staff - 5 years, 2 months ago

JEE Mains 2016 Mathematics 36

1 solution

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