Review question

# How many solutions are there to $(x^2+1)^{10} = 2x - x^2 - 2$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9754

## Solution

The equation $(x^2+1)^{10} = 2x - x^2 - 2$

1. has $x=2$ as a solution;

2. has no real solutions;

3. has an odd number of real solutions;

4. has twenty real solutions.

We can complete the square on the right-hand side of the equation to give $(x^2+1)^{10} = -1 - (x-1)^2.$

Now for real $x$, the left-hand side of this equation is always positive, but the right-hand side is always negative, so the equation has no real solutions.

The graphs of $y=(x^2+1)^{10}$ and $y= -1 - (x-1)^2$ help us to see what is going on.