- - 2. 2. Rozwiąż równanie. a) (x - 5)(x+5) = x2 – 100x b b) (3 – 1 )2 – (2x + 3) = ? ? c) 4(41 – 3)² = (6 – x) 2 - 2 9 d) 4(x + 2)2 - (2x - 1)2 = 200 + 10 e) (6 + 3x) (-5x + 6) + (x - 4) * = 4 f) (-4x – 3)(4x – 3)+8(1 - V2.0) = 1 3. Rozwiaż nierówność. Zaznacz na osi liczbowej zbiór rozwiazan

[tex]a)\\(x-5)(x+5)=x^2-100x\\x^2-5^2=x^2-100x\\x^2-25=x^2-100x\\x^2-x^2+100x=25\\100x=25 /:100\\x=\frac{25}{100}=\frac14\\\\b)\\(3-x)^2-(x+\frac13)^2=\frac29\\9-6x+x^2-(x^2+\frac23x+\frac19)=\frac29\\9-6x+x^2-x^2-\frac23x-\frac19=\frac29\\-6x-\frac23x=\frac29-9+\frac19\\-6\frac23x=\frac39-8\frac99\\-\frac{20}3x=-8\frac69\\-\frac{20}3x=-\frac{78}9 /*(-3)\\20x=26 /:20\\x=\frac{26}{20}=\frac{13}{10}=1.3[/tex]

[tex]c)\\4(\frac12x-3)^2=(6-x)^2\\4(\frac14x^2-3x+9)=36-12x+x^2\\x^2-12x+36=36-12x+x^2\\x^2-x^2-12x+12x+36-36=0\\0=0\\\\d)\\4(x+2)^2-(2x-1)^2=20x+10\\4(x^2+4x+4)-(4x^2-4x+1)=20x+10\\4x^2+16x+16-4x^2+4x-1=20x+10\\4x^2-4x^2+16x+4x-20x=10-16+1\\20x-20x=-5\\0\neq -5[/tex]

[tex]e)\\(6+\frac13x)(-\frac13x+6)+(\frac13x-4)^2=4\\(6+\frac13x)(6-\frac13x)+(\frac13x-4)^2=4\\36-\frac19x^2+(\frac19x^2-\frac83x+16)=4\\36-\frac19x^2+\frac19x^2-\frac83x+16=4\\-\frac19x^2+\frac19x^2-\frac83x=4-36-16\\\frac83x=-48 /*3\\8x=-144 /:8\\x=-18\\\\f)\\(-4x-3)(4x-3)+8(1-\sqrt2x)^2=1\\-(4x+3)(4x-3)+8(1-\sqrt2x)^2=1\\-(16x^2-9)+8(1-2\sqrt2x+2x^2)=1\\-16x^2+9+8-16\sqrt2x+16x^2=1\\-16x^2+16x^2-16\sqrt2x=1-9-8\\-16\sqrt2x=-16 /:(-16\sqrt2)\\[/tex]

[tex]x=\frac{16}{16\sqrt2}=\frac1{\sqrt2}=\frac{\sqrt2}2[/tex]

Magdaviz Magdaviz · Answer 2

Odpowiedź:

[tex]a)\\\\(x-5)(x+5)=x^2-100x\\\\x^2-25=x^2-100x\\\\x^2-x^2+100x=25\\\\100x=25\ \ /:100\\\\x=\frac{25}{100}\\\\x=\frac{1}{4}\\\\\\b)\\\\(3-x)^2-(x+\frac{1}{3})^2=\frac{2}{9}\\\\9-6x+x^2-(x^2+\frac{2}{3}x+\frac{1}{9})=\frac{2}{9}\\\\9-6x+x^2-x^2-\frac{2}{3}x-\frac{1}{9}=\frac{2}{9}\\\\9-6x-\frac{2}{3}x-\frac{1}{9}=\frac{2}{9}\ \ /\cdot9\\\\81-54x-6x-1=2\\\\80-60x=2\\\\-60x=2-80\\\\-60x=-78\ \ /:(-60)\\\\x=\frac{78}{60}\\\\x=\frac{13}{10}\\\\x=1\frac{3}{10}[/tex]

[tex]c)\\\\4(\frac{1}{2}x-3)^2=(6-x)^2\\\\4(\frac{1}{4}x^2-3x+9)=36-12x+x^2\\\\x^2-12x+36=36-12x+x^2\\\\x^2-12x+12x-x^2=36-36\\\\0=0\\\\R\'ownanie\ \ to\.zsamo\'sciowe\ \ ma\ \ niesko\'nczenie\ \ wiele\ \ rozwiaza\'n\\\\\\d)\\\\4(x+2)^2-(2x-1)^2=20x+10\\\\4(x^2+4x+4)-(4x^2-4x+1)=20x+10\\\\4x^2+16x+16-4x^2+4x-1=20x+10\\\\20x+15=20x+10\\\\20x-20x=10-15\\\\0=-5\\\\R\'ownanie\ \ sprzeczne\ \ nie\ \ ma\ \ rozwiazania[/tex]

[tex]e)\\\\(6+\frac{1}{3}x)(-\frac{1}{3}x+6)+(\frac{1}{3}x-4)^2=4\\\\(6+\frac{1}{3}x)(6-\frac{1}{3}x)+(\frac{1}{9}x^2-\frac{8}{3}x+16)=4\\\\6^2-(\frac{1}{3}x)^2+\frac{1}{9}x^2-\frac{8}{3}x+16=4\\\\36-\frac{1}{9}x^2+\frac{1}{9}x^2-\frac{8}{3}x+16=4\\\\52-\frac{8}{3}x=4\\\\-\frac{8}{3}x=4-52\\\\-\frac{8}{3}x=-48\ \ /\cdot3\\\\-8x=-144\ \ /:(-8)\\\\x=18[/tex]

[tex]f)\\\\(-4x-3)(4x-3)+8(1-\sqrt{2}x)^2=1\\\\-16x^2+12x-12x+9+8(1-2\sqrt{2}x+2x^2)=1\\\\-16x^2+9+8-16\sqrt{2}x+16x^2=1\\\\17-16\sqrt{2}x=1\\\\-16\sqrt{2}x=1-17\\\\-16\sqrt{2}x=-16\ \ /:(-16\sqrt{2})\\\\x=\frac{16}{16\sqrt{2}}\\\\x=\frac{1}{\sqrt{2}}\\\\x=\frac{1}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}\\\\x=\frac{\sqrt{2}}{2}[/tex]