Question

Questão 2: Simplifique as expressões
a) (x + 1)² + (x + 2)² - (2x + 1)²
b) (a² + b²)² - 2(ab)²
c) (x + 1)(x + 2) - 2(x + 2)² + (x + 2)(x + 3)​

Answer 1

Explicação passo-a-passo:

$a) \: (x + 1) ^{2} + (x + 2) ^{2} - (2x + 1) ^{2} \\ \\ {x}^{2} + 2x \times 1 + {1}^{2} + {x}^{2} + 2x \times 2 + {2}^{2} - ( {2x}^{2} + 2 \times 2x \times 1 + {1}^{2} ) = \\ \\ = {x}^{2} + 2x + 1 + {x}^{2} + 4x + 4 - (2 {x}^{2} + 4x + 1) \\ \\ = {x}^{2} + 2x + 1 + {x}^{2} + 4x + 4 - 2 {x}^{2} - 4x - 1 \\ \\ = {x}^{2} + {x}^{2} - {2x}^{2} + 2x + 4x - 4x + 1 + 4 - 1 \\ \\ = {2x}^{2} - {2x}^{2} + 6x - 4x + 5 - 1 \\ \\ = 2x + 4 \\ \\ \\ \\ b) \: ( {a}^{2} + {b}^{2} ) ^{2} - 2(ab)^{2} \\ \\ = ( {a}^{2} )^{2} + 2 {a}^{2} \times {b}^{2} + ( {b}^{2} )^{2} - 2 {a}^{2} {b}^{2} \\ \\ = {a}^{4} + {2a}^{2} {b}^{2} + {b}^{4} - {2a}^{2} {b}^{2} \\ \\ = {a}^{4} + {b}^{4} \\ \\ \\ \\ c) \: (x + 1)(x + 2) - 2(x + 2)^{2} + (x + 2)(x + 3) \\ \\ x \times x + x \times 2 + 1 \times x + 1 \times 2 - 2( {x}^{2} + 2x \times 2 + {2}^{2} ) + x \times x + x \times 3 + 2 \times x + 2 \times 3 \\ \\ = {x}^{2} + 2x + x + 2 - 2( {x}^{2} + 4x + 4) + {x}^{2} + 3x + 2x + 6 \\ \\ = {x}^{2} + 3x + 2 - {2x}^{2} - 8x - 8 + {x}^{2} + 5x + 6 \\ \\ {x}^{2} - {2x}^{2} + {x}^{2} + 3x - 8x + 5x + 2 - 8 + 6 \\ \\ = - {x }^{2} + {x}^{2} + 8x - 8x - 6 + 6 \\ \\ = 0$

Espero ter ajudado !