Question

x^2 = y^2 + z^2 

CMR: (5x - 3y + 4z)(5x-3y-4z) = (3x-5y)^2

 

Answer 1

Ta có:

\(\left(5x-3y+4z\right)\left(5x-3y-4z\right)\\ =\left[\left(5x-3y\right)+4z\right]\left[\left(5x-3y\right)-4z\right]\\ =\left(5x-3y\right)^2-\left(4z\right)^2\\ =25x^2-30xy+9y^2-16z^2\left(1\right)\)

Mà: \(x^2=y^2+z^2=>z^2=x^2-y^2\)

\(\left(1\right)=25x^2-30xy+9y^2-16\left(x^2-y^2\right)\\ =25x^2-30xy+9y^2-16x^2+16y^2\\ =9x^2-30xy+25y^2\\ =\left(3x\right)^2-2\cdot3x\cdot5y+\left(5y\right)^2\\ =\left(3x-5y\right)^2\)