Решите уравнение: 25.9. a) 3x² + 32x + 80 = 0; б) 100x2 - 160x + 63 = 0; в) 5x² + 26x - 24 = 0; г) 4x² - 12x + 9 = 0.

Решение:

1. 25.9. а) 3x² + 32x + 80 = 0




• \( a = 3, b = 32, c = 80 \)


• \( D = b^2 - 4ac = 32^2 - 4 · 3 · 80 = 1024 - 960 = 64 \)


• \( \sqrt{D} = 8 \)


• \( x_1 = \frac{-32 + 8}{2 · 3} = \frac{-24}{6} = -4 \)


• \( x_2 = \frac{-32 - 8}{2 · 3} = \frac{-40}{6} = -\frac{20}{3} \)




2. 25.9. б) 100x² - 160x + 63 = 0




• \( a = 100, b = -160, c = 63 \)


• \( D = b^2 - 4ac = (-160)^2 - 4 · 100 · 63 = 25600 - 25200 = 400 \)


• \( \sqrt{D} = 20 \)


• \( x_1 = \frac{160 + 20}{2 · 100} = \frac{180}{200} = \frac{9}{10} \)


• \( x_2 = \frac{160 - 20}{2 · 100} = \frac{140}{200} = \frac{7}{10} \)




3. 25.9. в) 5x² + 26x - 24 = 0




• \( a = 5, b = 26, c = -24 \)


• \( D = b^2 - 4ac = 26^2 - 4 · 5 · (-24) = 676 + 480 = 1156 \)


• \( \sqrt{D} = 34 \)


• \( x_1 = \frac{-26 + 34}{2 · 5} = \frac{8}{10} = \frac{4}{5} \)


• \( x_2 = \frac{-26 - 34}{2 · 5} = \frac{-60}{10} = -6 \)




4. 25.9. г) 4x² - 12x + 9 = 0




• \( a = 4, b = -12, c = 9 \)


• \( D = b^2 - 4ac = (-12)^2 - 4 · 4 · 9 = 144 - 144 = 0 \)


• \( x = \frac{-b}{2a} = \frac{12}{2 · 4} = \frac{12}{8} = \frac{3}{2} \)

Ответ: а) \( x_1 = -4, x_2 = -\frac{20}{3} \); б) \( x_1 = \frac{9}{10}, x_2 = \frac{7}{10} \); в) \( x_1 = \frac{4}{5}, x_2 = -6 \); г) \( x = \frac{3}{2} \).