Odpowiedź:
a)
[tex] {x}^{2} - 6x - 7 = 0 \\ \\ \Delta = {b}^{2} - 4ac \\ ( \Delta =( - 6 {)}^{2} - 4 \times 1 \times ( - 7) = 36 + 28 = 64\\ \: x_{1 }= \frac{ - b - \sqrt{ \Delta} }{2a} = \frac{6 - 8}{2} = \frac{ - 2}{2} = - 1 \\ x_{2} = \frac{ - b + \sqrt{ \Delta} }{2a} = \frac{6 + 8}{2} = \frac{ 14}{2} = 7[/tex]
b)
[tex] {x}^{2} - x + \frac{1}{4} = 0 \\ \Delta = ( - 1 {)}^{2} - 4 \times 1 \times \frac{1}{4} 1 - 1 = 0 \\ \\ x_{0 }= \frac{ - b}{2a} = \frac{1}{2} [/tex]
c)
[tex] {x}^{2} - 4x + 25 = 0 \\ \Delta= ( - 4 {)}^{2} - 4 \times 1 \times 25 = 16 - 100 = - 84[/tex]
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d)
[tex] - 2 {x}^{2} + x + 3 = 0 \\ \Delta = {1}^{2} - 4 \times ( - 2) \times 3 = + 24 = 25 \\ x_{1 }= \frac{ - 1 - 5}{ - 4} = - \frac{3}{2} \\ x_{2} = \frac{ - 1 + 5}{ - 4} = \frac{4}{ - 4} = - 1[/tex]
e)
[tex] { - x}^{2} - 20x - 75 = 0 \\ \Delta = ( - 20 {)}^{2} - 4 \times ( - 1) \times ( - 75) = 400 - 300 = 100 \\ x_{1 }= \frac{20 - 10}{ - 2} = \frac{10}{ - 2} = - 5 \\ x_{2 }= \frac{20 + 10}{ - 2} = \frac{30}{ - 2} = - 15[/tex]
f)
[tex]3 {z}^{2} - 2z + 10 = 0 \\ \Delta = ( - 2 {)}^{2} - 4 \times 3 \times 10 = 4 - 120 = - 116[/tex]
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