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Here's the solution to your math problem:

The problem asks to find the value of $x$ in the equation:
√(2x + 3) - √(x - 2) = 2

• Step 1: Isolate one of the square roots. Add √(x - 2) to both sides of the equation. √(2x + 3) = 2 + √(x - 2)

• Step 2: Square both sides of the equation to eliminate the square root on the left. (√(2x + 3))² = (2 + √(x - 2))² 2x + 3 = 2² + 2(2)√(x - 2) + (√(x - 2))² 2x + 3 = 4 + 4√(x - 2) + (x - 2)

• Step 3: Simplify the equation and isolate the remaining square root term. 2x + 3 = 4 + 4√(x - 2) + x - 2 2x + 3 = x + 2 + 4√(x - 2) Subtract $x$ and 2 from both sides: (2x - x) + (3 - 2) = 4√(x - 2) x + 1 = 4√(x - 2)

• Step 4: Square both sides of the equation again to eliminate the remaining square root. (x + 1)² = (4√(x - 2))² x² + 2x + 1 = 16(x - 2) x² + 2x + 1 = 16x - 32

• Step 5: Rearrange the equation into a standard quadratic form (ax² + bx + c = 0). x² + 2x + 1 - 16x + 32 = 0 x² - 14x + 33 = 0

• Step 6: Solve the quadratic equation. You can use factoring, the quadratic formula, or completing the square. Let's use factoring. We need two numbers that multiply to 33 and add to -14. These numbers are -3 and -11. (x - 3)(x - 11) = 0

• Step 7: Set each factor equal to zero and solve for $x$. x - 3 = 0 ⟹ x = 3 x - 11 = 0 ⟹ x = 11

• Step 8: Check both potential solutions in the original equation to ensure they are valid and not extraneous.

  • Check x = 3: √(2(3) + 3) - √(3 - 2) = √(6 + 3) - √1 = √9 - 1 = 3 - 1 = 2. This is correct.
  • Check x = 11: √(2(11) + 3) - √(11 - 2) = √(22 + 3) - √9 = √25 - 3 = 5 - 3 = 2. This is also correct.

Both values of $x$ satisfy the original equation.

The solutions are x = 3 and x = 11.