Not many solutions? Part 2

Algebra Level 3

Let the functions $f(x)$ and $g(x)$ be $f(x)=x^{3}-2x^{2}+x-2$ and $g(x)=x^{2}+1$ .

Find the sum of all real solutions to the equation

$f(x)+g(x)=0$ .

1 solution

Tarmo Taipale
Jun 18, 2016

$f(x)+g(x)=(x^{3}-2x^{2}+x-2)+(x^{2}+1)$ .

We get the equation

$x^{3}-2x^{2}+x-2+x^{2}+1=0$

$x^{3}-x^{2}+x-1=0$

$(x-1)(x^{2}+1)=0$

$x=1$ or $x^{2}+1=0$ . The second equation of these two has no real solutions, so we get

$x=1$ .

The only solution is 1, so the sum of all real solutions is also $\boxed{1}$ .

i and minus i works as well but their sum is 0 anyway

דניאל נוקראי - 4 years, 12 months ago

Well, I simply thought complex numbers don't count, except if it is said otherwise. But now I have edited the question to ask for real numbers.

Tarmo Taipale - 4 years, 12 months ago

Well then, the solutions should also be edited. Add "real" in front of "solutions"

Hung Woei Neoh - 4 years, 12 months ago

Tarmo Taipale - 4 years, 12 months ago