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2018 AMC 10A Problem 10

Problem 10 of 25EasierAlgebra

Suppose that real number xx satisfies 49−x2−25−x2=3.\sqrt{49-x^2}-\sqrt{25-x^2}=3. What is the value of 49−x2+25−x2?\sqrt{49-x^2}+\sqrt{25-x^2}?

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Solution

Note that the left hand side of the equation and the desired expression are conjugates. Multiplying them would remove the square roots. Multiplying them yields 49−x2−25+x2=24. 49 - x^2 - 25 + x^2 = 24. This means that the product of the values of the expressions is equal to 24.24. The desired value is therefore 24÷3=8.24 \div 3 = 8. Thus, A is the correct answer.

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Concepts: radical · difference of squares · algebraic manipulation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.