Bài 3. Hằng đẳng thức đáng nhớ

1: \(x\left(1-x\right)+\left(x-1\right)^2\)

\(=x-x^2+x^2-2x+1\)

=-x+1

2: \(\left(x-3\right)^2-x^2+10x-7\)

\(=x^2-6x+9-x^2+10x-7\)

=4x+9-7

=4x+2

3: \(\left(x+2\right)^2-\left(x-3\right)\left(x+1\right)\)

\(=x^2+4x+4-\left(x^2+x-3x-3\right)\)

\(=x^2+4x+4-x^2+2x+3=6x+7\)

4: \(\left(x+4\right)\left(x-2\right)-\left(x-3\right)^2\)

\(=x^2-2x+4x-8-\left(x^2-6x+9\right)\)

\(=x^2+2x-8-x^2+6x-9=8x-17\)

5: \(\left(x-2\right)^2+\left(x-1\right)\left(x+5\right)\)

\(=x^2-4x+4+x^2+5x-x-5\)

=\(2x^2-1\)

6: \(\left(x+3\right)\left(x-3\right)-x\left(x+23\right)\)

\(=x^2-9-x^2-23x\)

=-23x-9

7: \(\left(1-2x\right)\left(5-3x\right)+\left(4-x\right)^2\)

\(=5-3x-10x+6x^2+x^2-8x+16\)

\(=7x^2-21x+21\)

8: \(\left(x-2\right)\left(x+2\right)-\left(x-3\right)\left(x+1\right)\)

\(=x^2-4-\left(x^2+x-3x-3\right)\)

\(=x^2-4-\left(x^2-2x-3\right)\)

\(=x^2-4-x^2+2x+3=2x-1\)

9: \(\left(x+1\right)^2+\left(x-2\right)\left(x+2\right)-4x\)

\(=x^2+2x+1+x^2-4-4x\)

\(=2x^2-2x-3\)

10: \(\left(x+2\right)^2-\left(x+3\right)\left(x-3\right)+10\)

\(=x^2+4x+4-\left(x^2-9\right)+10\)

\(=x^2+4x+14-x^2+9=4x+23\)

11: \(\left(x+4\right)^2+\left(x+5\right)\left(x-5\right)-2x\left(x+1\right)\)

\(=x^2+8x+16+x^2-25-2x^2-2x\)

\(=8x-2x+16-25=6x-9\)

12: \(\left(x-1\right)^2-\left(x-4\right)\left(x+4\right)+\left(x+3\right)^2\)

\(=x^2-2x+1-\left(x^2-16\right)+x^2+6x+9\)

\(=2x^2+4x+10-x^2+16=x^2+4x+26\)

13: \(\left(x-1\right)^2-2\left(x+3\right)\left(x-3\right)+4x\left(x-4\right)\)

\(=x^2-2x+1-2\left(x^2-9\right)+4x^2-16x\)

\(=5x^2-18x+1-2x^2+18=3x^2-18x+19\)

14: \(\left(y-3\right)\left(y+3\right)\left(y^2+9\right)-\left(y^2+2\right)\left(y^2-2\right)\)

\(=\left(y^2-9\right)\left(y^2+9\right)-\left(y^4-4\right)\)

\(=y^4-81-y^4+4=-81+4=-77\)

\(9x^2-y^2\\ =\left(3x\right)^2-y^2\\ =\left(3x-y\right)\left(3x+y\right)\)

\(9x^2\) - \(y^2\) = \(\left(3x\right)^2\) - \(y^2\) = ( 3x + y )( 3x - y ) 

( Hằng đẳng thức Hiệu của hai bình phương )