Question

Oblicz wartość liczbową wyrażeń:
a) (a+1)²+(a-2)²-2(a-2)² dla a=5,5
b) (ax+1)²-(ax+1)(ax-1)+ax+1 dla a=⅓;x=√2
c) (1+2x)²-(1-2x)²-4(x+y) dla x=√3+√2;y=√3-√2
d) (x-2)²-(3x+2)²+2(2x+1)² dla x=-¼
e) 4(2x-y)²-x²+(x+2y)(x-2y) dla x=2√3;y=√3

Answer 1

A) (a+1)²+(a-2)²-2(a-2)² =a² + 2a + 1 + a² - 4a + 4 - 2(a² -4a +4) = a² + 2a + 1 + a² - 4a + 4 - 2a² +8a - 8 = 6a -3 = 6 x 5,5 - 3 = 30
dla a=5,5
b) (ax+1)²-(ax+1)(ax-1)+ax+1 = a²x²+ 2ax +1 - (a²x² -1) + ax +1 = a²x² +2ax +1 -a²x² +1 +ax +1 = 3ax + 3 = 3 x 1/3 x √2 + 3 = √2 + 3
dla a=⅓;x=√2
c) (1+2x)²-(1-2x)²-4(x+y) = 1 + 4x + 4x² - 1 +4x - 4x² -4x -4y = 4x - 4y = 4(√3 +√2) - 4(√3-√2) = 4√3 +4√2 -4√3 +4/2 = 8√2
dla x=√3+√2;y=√3-√2
d) (x-2)²-(3x+2)²+2(2x+1)² = x² - 4x +4 - 9x² - 12x -4 +2(4x² +4x + 1) = x² - 4x + 4 -9x² -12x -4 + 8x² +8x + 2 = -8x +2 = -8 x -1/4 +2 = 2 + 2= 4
dla x=-¼
e) 4(2x-y)²-x²+(x+2y)(x-2y) = 4(4x² - 4xy +y²) - x² + x² - 4y² = 16x² -16xy + 4y² -x² + x²-4y² = 16x² - 16xy = 16 x (2√3)² - 16 x 2√3 x √3 = 16 x 4 x 3 - 16 x 2 x 3 = 192 - 96 = 96
dla x=2√3;y=√3

Answer 2

A)(a+1)²+(a-2)²-2(a-2)²=a²+2a+1+(a²-4a+4)-2(a²-4a+4)=
a²+2a+1+a²-4a+4-2a²+8a-8=6a+13 6×5,5+13=46

b)(ax+1)²-(ax+1)(ax-1)+ax+1=a²x²+2ax+1-(a²x²-1)+ax+1=
a²x²+2ax+1-a²x²+1+ax+1=3ax+3 3×⅓×√2+3=√2+3

c)(1+2x)²-(1-2x)²-4(x+y)=1+4x+4x²-(1-4x+4x²)-4x-4y=
1+4x+4x²-1+4x-4x²-4x-4y=4x-4y 4(√3+√2)-4(√3-√2)=
4√3+4√2-4√3+4√2=4√2+4√2=8√2

d)(x-2)²-(3x+2)²+2(2x+1)²=x²-4x+2-(9x²+12x+4)+2(4x²+4x+1)=
x²-4x+2-9x²-12x-4+8x²+8x+2=-8x -8×(-¼)=2

e)4(2x-y)²-x²+(x+2y)(x-2y)=4(4x²-4xy+y²)-x²+x²-4y²=
16x²-16xy+4y²-x²+x²-4y²=16x²-16xy 16(2√3)²-16×2√3×√3 =16×12-16×6=
192-96=96