sqrt(x+2-4sqrt(x-2))+sqrt(x+7-6sqrt(x-2))=1

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Denote Square Root Terms: Let's denote the two square root terms as A and B, where A = √(x+2-4√(x-2)) and B = √(x+7-6√(x-2)). The equation can then be written as A + B = 1.Square Both Sides: We will square both sides of the equation to eliminate the square roots. This gives us (A + B)² = 1².Expand Left Side: Expanding the left side, we get A² + 2AB + B² = 1.Substitute Expressions: Substitute back the expressions for A and B to get (√(x+2-4√(x-2)))² + 2√(x+2-4√(x-2))√(x+7-6√(x-2)) + (√(x+7-6√(x-2)))² = 1.Simplify Squared Terms: Simplify the squared terms to get (x+2-4√(x-2)) + 2√((x+2-4√(x-2))(x+7-6√(x-2))) + (x+7-6√(x-2)) = 1.Combine Like Terms: Combine like terms to get x + x + 2 - 4√(x-2) + 7 - 6√(x-2) + 2√((x+2-4√(x-2))(x+7-6√(x-2))) = 1.Final Simplification: This simplifies to 2x + 9 - 10√(x-2) + 2√((x+2-4√(x-2))(x+7-6√(x-2))) = 1.Subtract and Divide: Subtract 9 from both sides to get 2x - 10√(x-2) + 2√((x+2-4√(x-2))(x+7-6√(x-2))) = -8.Isolate Square Roots: Divide the entire equation by 2 to simplify, yielding x - 5√(x-2) + √((x+2-4√(x-2))(x+7-6√(x-2))) = -4.Reconsider Strategy: Now, we need to simplify the term under the square root. However, this is a complex expression, and it seems like we might have made a mistake in our approach. The term under the square root does not simplify easily, and we should have looked for a way to isolate one of the square roots before squaring the equation. This indicates that we need to reconsider our strategy.

$\sqrt{x+2-4 \sqrt{x-2}}+\sqrt{x+7-6 \sqrt{x-2}}=1$

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x+2−4x−2+x+7−6x−2=1 \sqrt{x+2-4 \sqrt{x-2}}+\sqrt{x+7-6 \sqrt{x-2}}=1 x+2−4x−2​​+x+7−6x−2​​=1

$\sqrt{x+2-4 \sqrt{x-2}}+\sqrt{x+7-6 \sqrt{x-2}}=1$