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# Number Theory - AMC 10A, 2018 - Problem 10

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

AMC 10 A - 2018 - Problem No.10

Mathematics can be fun by Perelman

## Try with Hints

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

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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