[tex]1)\\\\( \frac{1}{2}x-3) (x+2) = (5-x) (6-x)\\\\\frac{1}{2}x^2+x-3x-6=30-5x-6x+x^2\\\\\frac{1}{2}x^2+x-3x-6-30+5x+6x-x^2 =0\\\\ -\frac{1}{2}x^2+9x -36 =0\\\\ a=-\frac{1}{2},\ \ \ b=9,\ \ c=-36\\\\\Delta =b^2-4ac=9^2-4*(-\frac{1}{2})*(-36)=81-72=9\\\\\sqrt{\Delta }=\sqrt{9}=3[/tex]
[tex]x_{1} =\frac{-b-\sqrt{\Delta }}{2a}=\frac{-9-3}{2*(-\frac{1}{2})} =\frac{-12}{-1} =12\\\\x_{2} =\frac{-b+\sqrt{\Delta }}{2a}=\frac{-9+3}{2*(-\frac{1}{2})} =\frac{-6}{-1} =6[/tex]
[tex]2)\\\\(x-3) (x+5) = (2x+3) (x-4)\\\\x^2+5x-3x-15=2x^2-8x+3x-12\\\\x^2+5x-3x-15-2x^2+8x-3x+12=0\\\\-x^2+7x-3=0 \\\\ a=-1 ,\ \ \ b=7,\ \ c=-3\\\\\Delta =b^2-4ac= 7^2-4*(-1)* (-3)= 49-12=37[/tex]
[tex]x_{1} =\frac{-b-\sqrt{\Delta }}{2a}=\frac{-7-\sqrt{37}}{2*(-1)} = \frac{-7-\sqrt{37}}{-2}= \frac{ 7+\sqrt{37}}{ 2} \\\\ x_{2} =\frac{-b+\sqrt{\Delta }}{2a}=\frac{-7+\sqrt{37}}{2*(-1)} = \frac{-7+\sqrt{37}}{-2}= \frac{ 7-\sqrt{37}}{ 2}[/tex]
[tex]3)\\\\(4x-1) (x-4)= (3x+2)(x-7)\\\\4x^2-16x-x+4=3x^2-21x+2x-14\\\\4x^2-16x-x+4-3x^2+21x-2x+14=0 \\\\ x^2+2x+18=0 \\\\ a= 1 ,\ \ \ b=2,\ \ c=18\\\\\Delta =b^2-4ac= 2^2-4*1*18= 4-72=-68\\\\\Delta < 0,\\\\brak\ rozwiazania[/tex]
