Thank you all for taking the time to help me.
Let's break the tie.
We want to simplify this:
[√(5 + 2√6) + √(5 - 2√6)]²
So let's expand that:
[√(5 + 2√6) + √(5 - 2√6)][√(5 + 2√6) + √(5 - 2√6)]
FOIL that:
√[(5 + 2√6)(5 + 2√6)] + √[(5 + 2√6)(5 - 2√6)] + √[(5 - 2√6)(5 + 2√6)] + √[(5 - 2√6)(5 - 2√6)]
The first and last terms become the square roots of squares so can be simplified. The middle two terms we can FOIL further within their own radicals:
√(5 + 2√6)² + √(25 - 10√6 + 10√6 - 4 * 6) + √(25 + 10√6 - 10√6 - 4 * 6) + √(5 - 2√6)²
The 10√6's all cancel, and the rest simplifies:
√(5 + 2√6)² + √(25 - 24) + √(25 - 24) + √(5 - 2√6)²
5 + 2√6 + √(1) + √(1) + 5 - 2√6
Now the 2√6's cancel, the √1's simplify, and the rest all add up together:
5 + √(1) + √(1) + 5
5 + 1 + 1 + 5
12
M^2 = (sqrt(5+2√6))^2 + (sqrt(5-2√6))^2 + (2)(sqrt(5+2√6))(sqrt(5-2√6))
= 5+2√6 + 5-2√6 + 2 (sqrt(5+2√6))(sqrt(5-2√6))
= 10 + (2) sqrt( (5+2√6) (5-2√6) )
= 10 + 2 sqrt( 25-10√6+10√6-4(6))
= 10 + 2 sqrt( 25-24)
= 10 + 2sqrt(1)
= 10+2
= 12
M^2=5+2sqr(6)+5-2sqr(6)+2sqr[(5+2sqr(6))(5-2sqr(6))]
=>
M^2=10+2sqr(25-24)
M^2=12
Use formula for square of binomial.
(a+b)² = a² + 2ab + b²
[ √(5 + 2√6) + √(5 - 2√6) ]²
= 5 + 2√6 + 2(5 + 2√6)(5 - 2√6) + 5 - 2√6
= 10 + 2(5 + 2√6)(5 - 2√6)
Difference of squares formula
(a-b)(a+b) = a² - b²
10 + 2(5 + 2√6)(5 - 2√6)
= 10 + 2(5² - (2√6)²)
= 10 + 2(25 - 4*6)
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Verified answer
Let's break the tie.
We want to simplify this:
[√(5 + 2√6) + √(5 - 2√6)]²
So let's expand that:
[√(5 + 2√6) + √(5 - 2√6)][√(5 + 2√6) + √(5 - 2√6)]
FOIL that:
√[(5 + 2√6)(5 + 2√6)] + √[(5 + 2√6)(5 - 2√6)] + √[(5 - 2√6)(5 + 2√6)] + √[(5 - 2√6)(5 - 2√6)]
The first and last terms become the square roots of squares so can be simplified. The middle two terms we can FOIL further within their own radicals:
√(5 + 2√6)² + √(25 - 10√6 + 10√6 - 4 * 6) + √(25 + 10√6 - 10√6 - 4 * 6) + √(5 - 2√6)²
The 10√6's all cancel, and the rest simplifies:
√(5 + 2√6)² + √(25 - 24) + √(25 - 24) + √(5 - 2√6)²
5 + 2√6 + √(1) + √(1) + 5 - 2√6
Now the 2√6's cancel, the √1's simplify, and the rest all add up together:
5 + √(1) + √(1) + 5
5 + 1 + 1 + 5
12
M^2 = (sqrt(5+2√6))^2 + (sqrt(5-2√6))^2 + (2)(sqrt(5+2√6))(sqrt(5-2√6))
= 5+2√6 + 5-2√6 + 2 (sqrt(5+2√6))(sqrt(5-2√6))
= 10 + (2) sqrt( (5+2√6) (5-2√6) )
= 10 + 2 sqrt( 25-10√6+10√6-4(6))
= 10 + 2 sqrt( 25-24)
= 10 + 2sqrt(1)
= 10+2
= 12
M^2=5+2sqr(6)+5-2sqr(6)+2sqr[(5+2sqr(6))(5-2sqr(6))]
=>
M^2=10+2sqr(25-24)
=>
M^2=12
Use formula for square of binomial.
(a+b)² = a² + 2ab + b²
[ √(5 + 2√6) + √(5 - 2√6) ]²
= 5 + 2√6 + 2(5 + 2√6)(5 - 2√6) + 5 - 2√6
= 10 + 2(5 + 2√6)(5 - 2√6)
Difference of squares formula
(a-b)(a+b) = a² - b²
10 + 2(5 + 2√6)(5 - 2√6)
= 10 + 2(5² - (2√6)²)
= 10 + 2(25 - 4*6)
= 12