Question

Zad.1.Rozwiąż równania. Na podstawie wzoru skróconego mnozenia

(a+b)2 = a2 + 2ab + b2.

a. (x+1)²-(x-3)²=x+1

b. (x+1)²+(x-3)(x+3)-2=2(x-1)²

c. (x+3)²-(4-x)(4+x)=2(x-1)²+1

d.(2-x)(2+x)+(x+3)²=x-7

Zad.2. Rozwiąż nierówność. Na podstawie wzoru skróconego mnozenia.

a. (x+4)²-(x+1)²≥ 4(x-1)

b.2(x-5)²-3(x+2)²<^-x²

c. (x-2)²-(x+5)(x-5)>-4(x+5)

d. 4(x-1)(x+1)-(2x-1)²>3

Answer 1

a) (x+1)²-(x-3)²=x+1

x²+2·x·1+1²-x²-2·x·3+3²=x+1

x²+2x+1-x²-6x+9=x+1

x²+2x-x²-6x-x=1-1          (x² redukuje sie)

2x-6x-x=0

-5x=0 / : (-5)

x=0

 

Answer 2

zad.1

a.(x+1)²-(x-3)²=x+1

(x²+2x+1)-(x²+6x+9)=x+1

x²+2x+1-x²-6x-9=x+1

-4x-8=x+1

-4x-x=1+8

-5x=9   /:(-5)

x=-9/5-ułamek

b.(x+1)²+(x-3)(x+3)-2=2(x-1)²

(x²+2x+1)+x²-9-2=2(x²-2x+1)

2x²+2x-10=2x²-4x+2

2x²+2x+4x-2x²=2+10

6x=12  /:6

x=2

c. (x+3)²-(4-x)(4+x)=2(x-1)²+1

(x²+6x+9)-(16-x²)=2(x²-2x+1)+1

x²+6x+9-16+x²=2x²-4x+2+1

2x²+6x-7=2x²-4x+3

2x²+6x-2x²+4x=3+7

10x=10  /:10

x=1

d.(2-x)(2+x)+(x+3)²=x-7

(4-x²)+(x²+6x+9)=x-7

4-x²+x²+6x+9=x-7

19=x-7

-x=-7-19

-x=-26  /:(-1)

x=26

zad.2

a. (x+4)²-(x+1)²≥ 4(x-1)

(x²+8x+16)-(x²+2x+1)≥4x-4

x²+8x+16-x²-2x-1≥4x-4

6x+15≥4x-4

6x-4x≥-4-15

2x≥-19  /:2

x≥-19/2-ułamek

b.2(x-5)²-3(x+2)²<-x²

2(x²-10x+25)-3(x²+4x+4)<-x²

2x²-20x+50-3x²-12x-12<-x²

-x²-32x+38<-x²

-x²+x²-32x<-38

-32x<-38  /:(-32)

x<38/32-ułamek

c. (x-2)²-(x+5)(x-5)>-4(x+5)

(x²-4x+4)-(x²-25)>-4x-20

x²-4x+4-x²+25>-4x-20

-4x+29>-4x-20

-4x-4x>-20-29

0>-49

d. 4(x-1)(x+1)-(2x-1)²>3

4(x²-1)-(4x²-4x+1)>3

4x²-4-4x²+4x-1>3

4x-5>3

4x>3+5

4x>8  /:4

x>2