Solve the following equation for x. x=◻sqrt(x^(2)-x-2)=x-2

Question

Solve the following equation for  x.  x=◻sqrt(x^(2)-x-2)=x-2

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Simplify by Squaring: Let's first simplify the equation by removing the square root. To do this, we will square both sides of the equation. x = √(x^2 - x - 2) Squaring both sides gives us: x^2 = (x^2 - x - 2)Move Terms and Set to Zero: Now, let's simplify the equation by moving all terms to one side to set the equation to zero. x^2 - (x^2 - x - 2) = 0 This simplifies to: x^2 - x^2 + x + 2 = 0Cancel x^2 Terms: After simplifying, we see that the x^2 terms cancel each other out, leaving us with: x + 2 = 0Solve for x: Now, we solve for x by subtracting 2 from both sides of the equation: x = -2Check Solution: We should check our solution by substituting x = -2 back into the original equation to ensure it satisfies the equation. Original equation: x = √(x^2 - x - 2) Substitute x = -2: -2 = √((-2)^2 - (-2) - 2) -2 = √(4 + 2 - 2) -2 = √(4) -2 = 2 or -2 = -2 We have an issue here because the square root of a number is non-negative, so -2 = 2 is not possible. The correct interpretation should be -2 = -2, which is true. However, we must remember that we squared the equation, which could have introduced an extraneous solution. We need to check the other part of the original equation: x = x - 2. Substitute x = -2: -2 = -2 - 2 -2 = -4 This is not true, so x = -2 is not a solution to the original equation.

Solve the following equation for $x$ . $\newline$ $\begin{array}{l} \sqrt{x^{2}-x-2}=x-2 \\ x=\square \end{array}$

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Solve the following equation for $x$ . $\newline$ $\begin{array}{l} \sqrt{x^{2}-x-2}=x-2 \\ x=\square \end{array}$