a)
[tex] 2{x}^{2} + 7x + 3 = 0[/tex]
∆=
[tex] {b}^{2} - 4ac = {7}^{2} - 4 \times 2 \times 3 = 49 - 24 = 25[/tex]
√∆=5
[tex] x_{1} = \frac{ - b - \sqrt{delty} }{2a} = \frac{ - 7 - 5}{2 \times 2} = \frac{ - 12}{4} = - 3[/tex]
[tex] x_{2} = \frac{ - b + \sqrt{delty} }{2a} = \frac{ - 7 + 5}{2 \times 2} = \frac{ -2}{4} = - \frac{1}{2} [/tex]
b)
[tex]4 {x}^{2} - x - 5 = 0[/tex]
∆=
[tex] {( - 1)}^{2} - 4 \times 4 \times ( - 5) = 1 - ( - 80) = 1 + 80 = 81[/tex]
√∆=9
[tex] x_{1} = \frac{ - ( - 1) - 9}{8} = \frac{1 - 9}{8} = \frac{ - 8}{8} = - 1[/tex]
[tex] x_{2} = \frac{ - ( - 1) + 9}{8} = \frac{1 + 9}{8} = \frac{10}{8} = 1 \frac{2}{8} = 1 \frac{1}{4} [/tex]
c)
[tex]3 {x}^{2} - 5x - 2 = 0[/tex]
∆=
[tex] {( - 5)}^{2} - 4 \times 3 \times ( - 2) = 25 - ( - 24) = 25 + 24 = 49[/tex]
√∆=7
[tex] x_{1} = \frac{ - ( - 5) - 7}{2 \times 3} = \frac{5 - 7}{6} = \frac{ - 2}{6} = - \frac{1}{3} [/tex]
[tex] x_{2} = \frac{ - ( - 5) + 7}{2 \times 3} = \frac{5 + 7}{6} = \frac{12}{6} = 6[/tex]
d)
[tex]28 {x}^{2} - 4x + \frac{1}{7} = 0[/tex]
∆=
[tex] {( - 4)}^{2} - 4 \times 28 \times \frac{1}{7} = 16 - 16 = 0[/tex]
∆=0
[tex] x_{0} = \frac{ - b}{2a} = \frac{ - ( - 4)}{2 \times 28} = \frac{4}{56} = \frac{1}{14} [/tex]
e)
[tex] - {x}^{2} + 6x + 1 = 0[/tex]
∆=
[tex] {6}^{2} - 4 \times ( - 1) \times 1 = 36 - ( - 4) = 36 + 4 = 40[/tex]
√∆=2√10
[tex] x_{1} = \frac{ - 6 - 2 \sqrt{10} }{ - 2} = 3 + \sqrt{10} [/tex]
[tex] x_{2} = \frac{ - 6 + 2 \sqrt{10} }{ - 2} = 3 - \sqrt{10} [/tex]
f)
[tex] \frac{1}{2}{x}^{2} - 2x - 1 = 0[/tex]
∆=
[tex] {( - 2)}^{2} - 4 \times \frac{1}{2} \times ( - 1) = 4 - ( - 2) = 4 + 2 = 6[/tex]
√∆=√6
[tex] x_{1} = \frac{ - ( - 2) - \sqrt{6} }{2 \times \frac{1}{2} } = 2 - \sqrt{6} [/tex]
[tex] x_{2} = \frac{ - ( - 2) + \sqrt{6} }{2 \times \frac{1}{2} } = 2 + \sqrt{6} [/tex]
g)
[tex] - 3 {x}^{2} + 2x - 3 = 0[/tex]
∆=
[tex] {2}^{2} - 4 \times (- 3) \times ( - 3) = 4 - 36 = - 32[/tex]
∆<0
Równanie nie ma rozwiązania.
h)
[tex]6{x}^{2} - 2x - 1 = 0[/tex]
∆=
[tex] {( - 2)}^{2} - 4 \times 6 \times ( -1 ) = 4 - ( - 24) = 4 + 24 = 28[/tex]
√∆=2√7
[tex] x_{1} = \frac{ - ( - 2) - 2 \sqrt{7} }{2 \times 6} = \frac{2 - 2 \sqrt{7} }{12} = \frac{1 - \sqrt{7} }{6} [/tex]
[tex] x_{2} = \frac{ - ( - 2) + 2 \sqrt{7} }{2 \times 6} = \frac{2 + 2 \sqrt{7} }{12} = \frac{1 + \sqrt{7} }{6} [/tex]
i)
[tex]2 {x}^{2} - \frac{5}{2}x + 1 = 0[/tex]
∆=
[tex] {( - \frac{5}{2}) }^{2} - 4 \times 2 \times 1 = \frac{25}{4} - 8 = 6 \frac{1}{4} - 8 = - 1 \frac{3}{4} [/tex]
∆<0
Równanie nie ma rozwiązania.
