Odpowiedź :
[tex]1)\\\\x^{2}-10x + 25 = 0\\\\(x-5)^{2} = 0 \ \ /\sqrt{}\\\\x-5 = 0\\\\\boxed{x_{o} = 5}[/tex]
[tex]2)\\\\-x^{2}+4x+21 = 0\\\\a = -1, \ b = 4, \ c = 21\\\\\Delta = b^{2}-4ac = 4^{2}-4\cdot(-1)\cdot21 = 16 +84 = 100\\\\\sqrt{\Delta} = \sqrt{100} = 10\\\\x_1 = \frac{-b+\sqrt{\Delta}}{2a} = \frac{-4+10}{2\cdot(-1)} =\frac{6}{-2} = -3\\\\x_2 = \frac{-b-\sqrt{\Delta}}{2a} = \frac{-4-10}{-2} = \frac{14}{2} = 7\\\\\boxed{x \in\{-3,7\}}[/tex]
[tex]3)\\\\2x^{2}+3x+8 = 0\\\\\Delta = 3^{2}-4\cdot2\cdot8} = 9 - 64 = -55\\\\\Delta < 0, \ brak \ rozwiazan[/tex]
[tex]4)\\\\9x^{2}-x = 0\\\\x(9x-1) = 0\\\\x = 0 \ \vee \ 9x-1 = 0\\\\x = 0 \ \vee \ 9x = 1 \ \ /:9\\\\x = 0 \ \vee \ x = \frac{1}{9}\\\\\boxed{x \in \{0,\frac {1}{9}\}}[/tex]
[tex]5)\\\\4x^{2}-1 = 0\\\\4x^{2}-1^{2} = 0\\\\(2x+1)(2x-1) = 0\\\\2x+1 = 0 \ \vee \ 2x-1 = 0\\\\2x = -1 \ \vee \ 2x = 1\\\\x = -\frac{1}{2} \ \vee \ x = \frac{1}{2}\\\\\boxed{x \in\{-\frac{1}{2}, \frac{1}{2}\}}[/tex]
Odpowiedź:
[tex]x=5[/tex]
[tex]x\in \{7,-3\}[/tex]
brak rozwiązań
[tex]x\in \{\frac{1}{9} ,0\}[/tex]
[tex]x\in \{\frac{1}{2}, -\frac{1}{2} \}[/tex]
Szczegółowe wyjaśnienie:
Przypomnijmy sobie wzór na obliczenie równania kwadratowego:
[tex]\Delta= b^{2} -4ac\\x_{1} =\frac{-b-\sqrt{\Delta} }{2a} \\x_{2} =\frac{-b+\sqrt{\Delta} }{2a}[/tex]
Rozwiązanie
[tex]x^{2} -10x+25\\ \Delta=10^{2} -4*1*25=100-100=0\\x=\frac{-(-10)}{2} =5[/tex]
[tex]-x^{2} +4x+21=0\\\Delta=4^{2} -4*(-1)*21=16+84=100\\x_{1} =\frac{-4-10}{-2} =\frac{-14}{-2} =7\\x_{2} =\frac{-4+10}{-2} =\frac{6}{-2} =-3\\[/tex]
[tex]2x^{2} +3x+8=0\\\Delta=3^{2} -4*2*8=9-64=-55\\\Delta<0[/tex]
brak rozwiązań
[tex]9x^{2} -x=0\\x(9x-1)=0\\x=0 \wedge 9x-1=0\\x=0 \wedge 9x=1\\x=0 \wedge x=\frac{1}{9}[/tex]
[tex]4x^{2} -1=0\\4x^{2} =1\\x^{2} =\frac{1}{4} \\x=\frac{1}{2} \wedge x=-\frac{1}{2}[/tex]
