Page 23 - Lösung Merkheft

Berechne!

a) (4x – 1)² = (x – 4)² b) (x – 4)² + (x – 3)² = 61

16x²-8x + 1 = x²-8x +16 |-1-x² x²-8x +16 +x²-6x +9 = 61 |-61

15x² = 15 |:15 2x² -14x -36 = 0

x² = 1 |√ −(−14) ± ඥ14 − 4 ∙ 2 ∙ (−36)
2
1,2 = 2 ∙ 2
x1 = + 1
14 ± ξ484 14 ± 22
x2 = -1 1,2 = 4 = 4

= 14 + 22 = 36 = 9
1
4 4
L = { 1 ; -1 } 14 − 22 −8
= 4 = 4 = −2
2

L = { 9 ; -2 }

c) (5x + 2)² = (3x + 4)² + 4(1 – x) d) (x + 5)² + (x + 3)² = 100

25x²+20x+4 = 9x²+24x +16+4-4x |-4-9x² x²+10x +25 +x²+6x +9 = 100 |-100

16x² = 16 |:16 2x² +16x -66 = 0

x² = 1 |√ −16 ± ඥ16 − 4 ∙ 2 ∙ (−66) 14 ± 28
2
1,2 = 2 ∙ 2 = 4
x1 = + 1
−16 + 28 12
= = = 3
1
x2 = -1 4 4
−16 − 28 −44
= 4 = 4 = −11
2
L = { 1 ; -1 } L = { 3 ; -11 }

e) (x + 5)(x – 5) = (2 – x)(2 + x) + 3 f) (x – 1)² – (2x – 2) = –(x – 3)²

x²-25 = 4 - x²+3 |+ 25 +x² x²-2x +1 -2x+2 = -x²+6x-9 |+x²-6x+9

2x² = 32 |:2 2x²-10x +12 = 0

x² = 16 |√ = −(−10)±ඥ10 −4∙2∙12 10±ξ4 = 1±2
2
=
1,2
2∙2 4 4
x1 = + 4 10 + 2 12
= = = 3
1
x2 = - 4 4 4
10 − 2 8
= = = 2
2
4 4
L = { 4 ; -4 } L = { 3 ; 2 } x² – 5x +6 = 0; L = { 2; 3 } f) 2x²=32; L = { 4; –4 } e)

x² + 8x – 33 = 0; L = { 3; –11 } d) 6x²=16; L = { 1; –1 } 15x²=15; L = { 1; –1 } b) a)

b) x² – 7x – 18 = 0; L = { 9; –2 }
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