[tex]E(x) = (x - 1) {}^{2} + (2x - 1) {}^{2} + (1 - x)(2x - 1) + 3x - 1[/tex]
[tex]E(x) = x {}^{2} - 2x + 1 + 4x {}^{2} - 4x + 1 + 2x - 1 - 2x {}^{2} + x + 3x - 1[/tex]
[tex]E(x) = x {}^{2} + 4x {}^{2} - 4x - 2x {}^{2} + x + 3x \\ E(x) = 3x {}^{2} \\ \\ \\ b) n = E(3 - \sqrt{3} ) + 2 \sqrt{243} \\ n = 3x {}^{2} + 2 \sqrt{243} \\ n = 3(3 - \sqrt{3} ) {}^{2} + 2 \sqrt{243} \\ n = 3(9 - 6 \sqrt{3} + 3) + 2 \times 9 \sqrt{3} \\ n = 3(12 - 6 \sqrt{3} ) + 18 \sqrt{3} \\ n = 36 - 18 \sqrt{3} + 18 \sqrt{3} \\ n = 36 \\ \iff \: n = 6 {}^{2} = p.p.[/tex]
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[tex]E(x) = (x - 1) {}^{2} + (2x - 1) {}^{2} + (1 - x)(2x - 1) + 3x - 1[/tex]
[tex]E(x) = x {}^{2} - 2x + 1 + 4x {}^{2} - 4x + 1 + 2x - 1 - 2x {}^{2} + x + 3x - 1[/tex]
[tex]E(x) = x {}^{2} + 4x {}^{2} - 4x - 2x {}^{2} + x + 3x \\ E(x) = 3x {}^{2} \\ \\ \\ b) n = E(3 - \sqrt{3} ) + 2 \sqrt{243} \\ n = 3x {}^{2} + 2 \sqrt{243} \\ n = 3(3 - \sqrt{3} ) {}^{2} + 2 \sqrt{243} \\ n = 3(9 - 6 \sqrt{3} + 3) + 2 \times 9 \sqrt{3} \\ n = 3(12 - 6 \sqrt{3} ) + 18 \sqrt{3} \\ n = 36 - 18 \sqrt{3} + 18 \sqrt{3} \\ n = 36 \\ \iff \: n = 6 {}^{2} = p.p.[/tex]