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Which value for the constant cc makes x=cx = c an extraneous solution in the following equation?

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Q. Which value for the constant cc makes x=cx = c an extraneous solution in the following equation?
  1. Identify Equation and Constant: Identify the equation and the constant involved.\newlineWe need the equation to find the constant that causes an extraneous solution.
  2. Set Up Analysis Equation: Set up the equation for analysis.\newlineAssume the equation is x+c=x+4 x + c = \sqrt{x + 4} , where c c is the constant.
  3. Square to Eliminate Root: Square both sides to eliminate the square root.\newline(x+c)2=x+4 (x + c)^2 = x + 4
  4. Expand and Simplify: Expand and simplify the squared expression.\newlinex2+2cx+c2=x+4 x^2 + 2cx + c^2 = x + 4
  5. Rearrange to Zero: Rearrange the equation to set it to zero.\newlinex2+2cx+c2−x−4=0 x^2 + 2cx + c^2 - x - 4 = 0
  6. Solve for x: Solve for x x using the quadratic formula.\newlinex=−2c+1±(2c−1)2−4(c2−4)2 x = \frac{-2c + 1 \pm \sqrt{(2c-1)^2 - 4(c^2-4)}}{2}

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