3.31
a)
[tex]4 {x}^{2} - 8x = 0 \\ x(4x - 8) = 0 \\ x = 0 \\i \\ 4x - 8 = 0 \\ 4x = 8 \\ x = 2[/tex]
b)
[tex] - 4 {x}^{2} + 20 = 0 \\ delta = {0}^{2} - 4 \times ( - 4) \times 20 = 320 \\ x1 = \frac{ - \sqrt{320} }{ - 8} = \frac{ - 4 \sqrt{20} }{ - 8} = \frac{1}{2} \sqrt{20} \\ x2 = \frac{ \sqrt{320} }{ - 8} = - \frac{1}{2} \sqrt{20} [/tex]
c)
[tex] \frac{1}{2} {x}^{2} + 12x = 0 \\ x( \frac{1}{2} x + 12) = 0 \\ x = 0 \\ x = - 24[/tex]
d)
[tex] {x}^{2} + 10x + 25 = 0 \\ delta = 100 - 100 = 0 \\ x = - 5[/tex]
e)
[tex]\frac{4}{5} {x}^{2} + 5 = 0 \\ delta = {0}^{2} - 4 \times \frac{4}{5} \times 5 < 0 \\ brak \: rozwiazan[/tex]
f)
[tex] - {x}^{2} + 2x - 1 = 0 \\ delta = 4 - 4 \times ( - 1) \times ( - 1) = 0 \\ x1 = \frac{ - 2}{ - 2} = 1[/tex]
3.32
a)
[tex] {9x}^{2} + 6x + 1 = 0 \\ d = 36 - 4 \times 9 \times 1 = 0 \\ x1 = \frac{ - 6}{18} = - \frac{1}{3} [/tex]
b)
[tex] \frac{3}{4} {x}^{2} - 3x =0 \\ x( \frac{3}{4} x - 3) = 0 \\ x1 = 0 \\ x2 = 4[/tex]
c)
[tex](5x - 4) {}^{2} = 0 \\ 25 {x}^{2} - 40x + 16 = 0 \\ d = 1600 - 1600 = 0 \\ x1 = \frac{40}{50} = \frac{4}{5} [/tex]
d)
[tex] - (x + 1) {}^{2} - 9 = 0 \\ - (x {}^{2} + 2x + 1) - 9 = 0 \\ - {x}^{2} - 2x - 10 = 0 \\ d = 4 - 4 \times ( - 1) \times ( - 10) = \\ = 4 - 40 < 0 \\ brak \: rozwiazan[/tex]
e)
[tex] \frac{1}{2} {x}^{2} + 8x + 32 = 0 \\ d = 64 - 64 = 0 \\ x1 = \frac{ - 8}{1} \\ x1 = - 8[/tex]
f)
[tex] \frac{3}{4 } {x}^{2} - 3x + 3 = 0 \\ d = 9 - 9 = 0 \\ x1 = 3 \div \frac{6}{4} = 3 \times \frac{4}{6} = 2[/tex]
