Question

Daca √(24-x²) - √(8-x²) = 2, atunci √(24-x²) + √(8-x²) este:
a)8; b)6; c)4; d)10; e)16.

Answer 1

$\displaystyle\\ \sqrt{24-x^2} - \sqrt{8-x^2} = 2\\\\ \sqrt{24-x^2} + \sqrt{8-x^2} = y\\\\ \text{Inmultim cele 2 ecuatii:}\\\\ \Big(\sqrt{24-x^2} - \sqrt{8-x^2}\Big)\Big(\sqrt{24-x^2} + \sqrt{8-x^2}\Big)=2y\\\\ \Big(\sqrt{24-x^2}\Big)^2 - \Big(\sqrt{8-x^2}\Big)^2 =2y\\\\ \Big(24-x^2\Big) - \Big(8-x^2\Big) =2y\\\\ 24-x^2 - 8+x^2 =2y\\\\ 24-8 -x^2 +x^2 =2y\\\\ 24-8 =2y\\\\ 16=2y\\\\ y = \frac{16}{2} =8\\\\ \Longrightarrow~~\boxed{\sqrt{24-x^2} + \sqrt{8-x^2}=8}\\\\ \text{Raspuns corect: }~\boxed{a)}$