Answer:
x = 1
Explanation:
You want the solution to the equation √(x²+2x+6) +x² = √(2x+2) -x +3.
Graphing calculator
We find a graphing calculator to be a useful tool for finding the solutions to equations like this. It shows the solution to be x = 1.
Solved in the usual way, this would resolve to an 8th degree equation with one integer factor: (x -1). The remaining 7th degree polynomial factor has 3 real roots and 4 complex roots. All of these 7 irrational solutions are extraneous. (They do not satisfy the original equation.)
The solution is x = 1.
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Additional comment
Our approach to removing the radicals is to ...
- subtract x², and square both sides
- subtract 11 -4x -5x² +2x³ +x⁴, and square both sides
- subtract the right side, leaving
-47 -84x +64x² +116x³ -2x⁴ -44x⁵ -8x⁶ +4x⁷ +x⁸ = 0
The usual approach to looking for roots is to use Descartes' rule of signs and the Rational Root theorem.
You can determine there are 3 or 1 positive real roots, and 5, 3, or 1 negative real roots. The only rational real roots will be ±1 or ±47, and we know that ±47 can't work. (47^8 will not be balanced by any of the other terms in the polynomial.)
You can see already that the graphing calculator approach is much preferred.
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