Question

If \(2 \sqrt{x}=x-3\), which of the following is the solution set for \(x\) ? A) \(\\{-1,9\\}\) B) \(\\{1,-9\\}\) C) \(\\{9\\}\) D) \(\\{1,9\\}\)

Short answer

The correct solution set for the given equation is C) \(\{9\}\).

Step-by-step solution

Square both sides of the equation

To eliminate the square root, square both sides of the equation \(2 \sqrt{x}=x-3\). This will give the equation: \[(2\sqrt{x})^2 = (x-3)^2.\]

Simplify the equation

Simplify both sides of the equation from the previous step. On the left side, the square and the square root will cancel out: \[(2\sqrt{x})^2 = 4x,\] while on the right side, expand the binomial: \[(x-3)^2 = x^2 - 6x + 9.\] This results in the equation: \[4x = x^2 - 6x + 9.\]

Rearrange and Solve the Quadratic Equation

To solve for x, move all terms to one side of the equation, creating a quadratic equation: \[x^2 - 10x + 9 = 0.\] Now, attempt to factor this quadratic equation: \[(x - 1)(x - 9)= 0\] The solutions for x are 1 and 9: \[x=1, x=9.\]

Verify the solutions

To verify whether the solutions are valid, plug each solution back into the original equation: For x = 1: \(2 \sqrt{1}=1-3\) gives \(2 \neq -2\). In this case, \(x=1\) is not a valid solution. For x = 9: \(2 \sqrt{9}=9-3\) gives \(6=6\). In this case, \(x=9\) is a valid solution. So, the correct solution set for the given equation is \(\{9\}\). The correct answer is: C) \(\{9\}\).