Answer:
The values of [tex]a[/tex] and [tex]b[/tex] are [tex]33[/tex] and [tex]\frac{20\sqrt{3}}{3}[/tex], respectively.
Step-by-step explanation:
The statement is equivalent to the following mathematic expression:
[tex]\left(5 + 2\sqrt{2})^{2} = a + b\cdot \sqrt{6}[/tex] (1)
By definition of the perfect square trinomial:
[tex]25 + 20\cdot \sqrt{2} + 8 = a + b\cdot \sqrt{6}[/tex]
[tex]33 + 20\sqrt{2} = a + b\cdot \sqrt{6}[/tex]
And by direct comparison we have the following system:
[tex]a = 33[/tex] (2)
[tex]b\cdot \sqrt{6} = 20\sqrt{2}[/tex] (3)
By (3), we solve for [tex]b[/tex]:
[tex]b = \frac{20}{\sqrt{3}}[/tex]
[tex]b = \frac{20\sqrt{3}}{3}[/tex]
The values of [tex]a[/tex] and [tex]b[/tex] are [tex]33[/tex] and [tex]\frac{20\sqrt{3}}{3}[/tex], respectively.
