Try this beautiful problem from AMC 10A, 2018 based on 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 $\sqrt {49-x^2}$ + $\sqrt {25-x^2}$?
Number Theory
Real number
Square root
Answer: 8
AMC 10 A - 2018 - Problem No.10
Mathematics can be fun by Perelman
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.
$\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\left(\sqrt{49-x^2}\right)-\left(\sqrt{25-x^2}\right)$
$=49-x^2-25+x^2=24$
Given that $\left.\sqrt{49-x^2}\right)-\left(\sqrt{25-x^2}\right)=3$
$\sqrt{49-x^2}+\sqrt{25-x^2}=\frac{24}{3} \quad=8$
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