Do you know its property?

Algebra Level 3

If $x^3, x, x^2$ form an arithmetic progression (in the given order), find the sum of all possible values of $x$ .

Clarification : An arithmetic progression is allowed to be a constant sequence.

In the answer options, $\phi = \frac{ 1 + \sqrt{5} } { 2}$ (the golden ratio).

\phi

1-\phi

1+\phi

-\phi

-1 2 0

3 solutions

Vishwak Srinivasan
Aug 8, 2015

If $x^3,x \text{ \& } x^2$ form an Arithmetic Sequence, then:

$2x = x^3 + x^2 \Rightarrow x^3 + x^2 - 2x = 0$

$x(x^2 + x - 2) = 0 \Rightarrow x(x+2)(x-1) = 0$

Now all the roots are distinct, which was the intention of bringing it this far(Vieta's could have been used and the answer comes out to be $-1$ ).

Hence the answer is $-2 + 1 + 0 = \boxed{-1}$ .

What's vieta I don't understand please elaborate

Uttkarsh Singh - 4 years, 8 months ago

It is a the formula for sum and Product of quadratic roots

Aashay Godbole - 6 months, 2 weeks ago

Caleb Townsend
Apr 9, 2015

$x-x^3 = x^2 - x \\ x^3 + x^2 - 2x = 0$ By Vieta's formula, the sum of the roots is $-\frac{1}{1} = \boxed{-1}$

FYI You should check that the roots are distinct (and possibly real?).

Calvin Lin Staff - 6 years, 2 months ago

How is -1,-1,1 an arithmetic progression?

Ian Dowler - 3 years, 10 months ago

Note that the question says "find the sum of all possible values". It does not say "what must x be".

The values are $x = 1, x = 0, x = - 2$ . You can verify that we get 3 arithmetic progressions from here.

Calvin Lin Staff - 3 years, 10 months ago

Ian, x = -1 is not one of the possible values for x; hope this clears things up. Regards, Ed Gray

Edwin Gray - 2 years, 9 months ago

Edwin Gray
Sep 13, 2018

Let x^3 = a, x = a + d, x^2 = a + 2d. The 2x - x^2 = x^3. Clearly x could be 0. if not divide by x, giving x^2 + x _ 2 = 0 or (x + 2)(x - 1) = 0, so x = -2, or x = 1. Substitution reveals that these values do produce an A. P. so the answer is 0 + 1 + (-2) = -1. Ed Gray

Do you know its property?

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