Try this beautiful problem from AMC 10A, 2018 based on Number theory.

Problem – Number Theory


Let’s try this problem number 10 from AMC 10A, 2018 based on Number Theory.

Suppose that the real number $x$ satisfies $\sqrt {49-x^2}$ – $\sqrt {25-x^2}$ = $3$.

What is the value of $\sqrt {49-x^2}$ + $\sqrt {25-x^2}$?

  • 8
  • $\sqrt 33 + 8$
  • 9
  • $2\sqrt10+4$
  • 12

Key Concepts


Number Theory

Real number

Square root

Check the Answer


But try the problem first…

Answer: 8

Source
Suggested Reading

AMC 10 A – 2018 – Problem No.10

Mathematics can be fun by Perelman

Try with Hints


First hint

As a first hint we can start from here :

In order to get rid of the square roots, we multiply by the conjugate. Its value is the solution.The \(x^2\) terms cancel out.

\((\sqrt {49 – x^2} +\sqrt {25 – x^2}) (\sqrt {49 – x^2}) -(\sqrt {25 – x^2})\)

= 49 -\(x^2 – 25 + x^2\)

=24

Final Hint

Given that \(\sqrt {49 – x^2}) -(\sqrt {25 – x^2})\) = 3

\(\sqrt {49 – x^2} +\sqrt {25 – x^2}\) = \(\frac {24}{3}\)

= 8

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