$(4+\sqrt{15})^x+(4-\sqrt{15})^x$

Algebra Level 4

What is the sum of all the $x$ 's that satisfy the equation $(4+\sqrt{15})^x+(4-\sqrt{15})^x=8?$

4 1 0 2

4 solutions

Rama Devi
May 24, 2015

The possible values of x for this equation are 1 and -1. Thus 1 + (-1) = 0

Brilliant really, thanks for help

Szalony Kierowca - 1 year, 7 months ago

Smit Kiri
Apr 20, 2014

The possible values of x for this equation are 1 and -1. Thus 1 + (-1) = 0

Outstanding, this is what I call explanation

Szalony Kierowca - 1 year, 7 months ago

Ramasubramaniyan Gunasridharan
Apr 19, 2014

x = 1 or -1. Thus 1-1 =0

Wow clown 🤡 pure genius

Szalony Kierowca - 1 year, 7 months ago

Finn Hulse
Apr 19, 2014

If $x=1$ the equation will obviously work. The real test is to find the other solution. I found it by accident. The other value for $x$ is $-1$ , which you can verify for yourself. Thus the desired sum is $\boxed{0}$ .

How did you know that there was another solution? and only one other solution?

milind prabhu - 6 years, 4 months ago

See this post (my answer) where a similar question was answered. Just change the power $x/3$ in my answer to $x$ and it would work for this problem.

The shorter approach is the establish the symmetry in the equation which is also explained by me in the comments of that post.

The intuition to solve the problem in this particular way comes from noting that $4^2-(\sqrt 5)^2=1$ .

Prasun Biswas - 5 years, 10 months ago

( 4 + 15 ) x + ( 4 − 15 ) x (4+\sqrt{15})^x+(4-\sqrt{15})^x ( 4 + 1 5 ​ ) x + ( 4 − 1 5 ​ ) x

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$(4+\sqrt{15})^x+(4-\sqrt{15})^x$