Question

1. Numarul a = (2x + 1)(2x + 5) - 16x - 4, x ∈ Z, este patratul perfect al numarului : ........ .
2. Valoarea numarului natural x = $\sqrt{11-6 \sqrt{2} } - \sqrt{9+4 \sqrt{2} } + \sqrt{3+2 \sqrt{2}}$ este egal cu : ....................... .

Answer 1

a=(2x+1)(2x+5)-16x-4=
=4x²+2x+10x+5-16x-4=
=4x²-4x+1=(2x-1)²

2.√(11-6√2)-√(9+4√2)+√(3+2√2)=
=√(11-√72)-√(9+√32)+√(3+√8)=3-√2-2√2-1+√2+1=3-2√2

√(11-√72)=√[11+√(121-72)]/2-√[(11-√(121-72)]/2=√(11+7)/2-√(11-7)/2=√9-√2=3-√2
√(9+√32)=√[9+√(81-32)]/2+√[(9-√(81-32)]/2=√(9+7)/2+√(9-7)/2=√8+1=2√2+1
√(3+√8)=√[3+√(9-8)]/2+√[(3-√(9-8)]/2=√(3+1)/2+√(3-1)/2=√2+1=√2+1
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2.√(11+6√2)-√(9+4√2)+√(3+2√2)=
=√(11+√72)-√(9+√32)+√(3+√8)=3+√2-2√2-1+√2+1=3

√(11+√72)=√[11+√(121-72)]/2+√[(11-√(121-72)]/2=√(11+7)/2+√(11-7)/2=
=√9- √2=3+√2
√(9+√32)=√[9+√(81-32)]/2+√[(9-√(81-32)]/2=√(9+7)/2+√(9-7)/2=√8+1=2√2+1
√(3+√8)=√[3+√(9-8)]/2+√[(3-√(9-8)]/2=√(3+1)/2+√(3-1)/2=√2+1=√2+1