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Solve the following equation for
x.

x=◻sqrt(x^(2)-x-2)=x-2

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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Find derivatives using the chain rule II

Full solution

Q. Solve the following equation for

x

.

\newline

\begin{array}{l} \sqrt{x^{2}-x-2}=x-2 \\ x=\square \end{array}

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. $\newline$ $x = \sqrt{x^2 - x - 2}$ $\newline$ Squaring both sides gives us: $\newline$ $x^2 = (x^2 - x - 2)$
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. $\newline$ $x^2 - (x^2 - x - 2) = 0$ $\newline$ This simplifies to: $\newline$ $x^2 - x^2 + x + 2 = 0$
Cancel $x^2$ Terms: After simplifying, we see that the $x^2$ terms cancel each other out, leaving us with: $x + 2 = 0$
Solve for x: Now, we solve for x by subtracting $2$ from both sides of the equation: $\newline$ $x = -2$
Check Solution: We should check our solution by substituting $x = -2$ back into the original equation to ensure it satisfies the equation. $\newline$ Original equation: $x = \sqrt{x^2 - x - 2}$ $\newline$ Substitute $x = -2$ : $\newline$ $-2 = \sqrt{(-2)^2 - (-2) - 2}$ $\newline$ $-2 = \sqrt{4 + 2 - 2}$ $\newline$ $-2 = \sqrt{4}$ $\newline$ $-2 = 2$ or $-2 = -2$ $\newline$ 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: $x = \sqrt{x^2 - x - 2}$ $0$ . $\newline$ Substitute $x = -2$ : $\newline$ $x = \sqrt{x^2 - x - 2}$ $2$ $\newline$ $x = \sqrt{x^2 - x - 2}$ $3$ $\newline$ This is not true, so $x = -2$ is not a solution to the original equation.

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