Solve for X

Algebra Level 3

Let all the solutions of $x$ satisfying the equation $(x^2-7x+11)^{x^2-17x +72} = 1$ be $x_1, x_2, \ldots, x_n$ , where $x_1<x_2<\ldots<x_n$ .

Submit your answer as the concatenation of these numbers, $\overline{x_1 x_2 \cdots x_n}$ .

For example, if you think that all the solutions of $x$ is 1, 6 and 7, then the answer is 167.

The answer is 234589.

1 solution

Srinivasa Gopal
Jan 5, 2019

If a (power) b is equal to 1 there are three cases where this situation is possible

a^b =1 when (a =1) or (b = 0 )or ( a = -1 and b is even)

Solving for these three cases yields Case 1 :

x^2 - 7x + 11 = 1 ; x^2 - 7x + 10 = 0; (x-2)(x-5) = 0; x= 2 or x= 5;

Case 2 : x^2 - 17x + 72 = 0 (x-8)(x-9) = 0 x=8 or x = 9

Case 3 : x^2 -7x + 11 = -1 x^2 -7x + 12 = 0 (x-3)(x-4) = 0 x= 3 or x = 4

Here it has to be checked if x^2 - 17x + 72 is even, when x = 3 the expression x^2 - 17x + 72 evaluates to 30 ,when x = 4 the expression x^2 - 17x + 72 evaluates to 20. Both times x^2 -7x + 11 raised to even powers will yield 1.

So the possible values of x which satisfy the above equation are 2,5,8,9,3,4, writing this in ascending order yields 234589

Just a suggestion, but just putting $\($ at the ends of math equations etc. formats them as $\LaTeX$ , which is better to look at/understand.

Parth Sankhe - 2 years, 5 months ago

Solve for X

The answer is 234589.

1 solution

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