Math Problem Statement

\sqrt{x}-\sqrt{x-2}=1

Solution

To solve the equation xx2=1\sqrt{x} - \sqrt{x-2} = 1, follow these steps:

  1. Isolate one of the square root terms:

    x=1+x2\sqrt{x} = 1 + \sqrt{x-2}
  2. Square both sides to eliminate the square root on the left:

    (x)2=(1+x2)2(\sqrt{x})^2 = (1 + \sqrt{x-2})^2
  3. Expand the right-hand side:

    x=1+2x2+(x2)x = 1 + 2\sqrt{x-2} + (x-2)

    Simplify the equation:

    x=x1+2x2x = x - 1 + 2\sqrt{x-2}
  4. Subtract xx from both sides:

    0=1+2x20 = -1 + 2\sqrt{x-2}
  5. Isolate the remaining square root term:

    1=2x21 = 2\sqrt{x-2}
  6. Divide both sides by 2:

    12=x2\frac{1}{2} = \sqrt{x-2}
  7. Square both sides again to eliminate the square root:

    (12)2=x2\left(\frac{1}{2}\right)^2 = x - 2 14=x2\frac{1}{4} = x - 2
  8. Solve for xx:

    x=2+14x = 2 + \frac{1}{4} x=84+14x = \frac{8}{4} + \frac{1}{4} x=94x = \frac{9}{4}

The solution to the equation xx2=1\sqrt{x} - \sqrt{x-2} = 1 is:

x=94x = \frac{9}{4}

To verify, substitute x=94x = \frac{9}{4} back into the original equation:

94942=3214=3212=1\sqrt{\frac{9}{4}} - \sqrt{\frac{9}{4} - 2} = \frac{3}{2} - \sqrt{\frac{1}{4}} = \frac{3}{2} - \frac{1}{2} = 1

The solution x=94x = \frac{9}{4} satisfies the original equation.

Would you like more details or have any questions?

Here are some related questions:

  1. How do you solve equations involving multiple square roots?
  2. What are the properties of square roots used in solving equations?
  3. Can you solve x+3x1=2\sqrt{x+3} - \sqrt{x-1} = 2?
  4. How do you verify solutions to square root equations?
  5. What are common mistakes to avoid when solving square root equations?
  6. How can you graph the function y=xx2y = \sqrt{x} - \sqrt{x-2}?
  7. What is the domain of the function y=xx2y = \sqrt{x} - \sqrt{x-2}?
  8. Can you solve 2x+1x=1\sqrt{2x+1} - \sqrt{x} = 1?

Tip: Always verify your solutions by substituting them back into the original equation to check if they satisfy the equation.

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Math Problem Analysis

Mathematical Concepts

Square Roots
Equations

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12