Question

Решите через дискриминат 14х²-5х-19=0 у²+3у-108=0

Answer 1

14х² - 5х - 19 = 0

a = 14, b = -5, c = -19

D = b² - 4ac = (-5)² - 4 * 14 * (-19) = 25 + 1 064 = 1089 = 33²

x₁₂ = -b ± $\sqrt{D}$ / 2a = -(-5) ± $\sqrt{1089}$ / 2 * 14 = 5 ± 33 / 28

x₁ = 5 + 33 / 28 = $\frac{38}{28}$

x₂ = 5 - 33 / 28 = $- \frac{28}{28}$ = -1

x₁ =  $\frac{38}{28}$

x₂ = -1

у² + 3у - 108 = 0

a = 1, b = 3, c = -108

D = b² - 4ac = 3² - 4 * 1 * (-108) = 9 + 432 = 441 = 21²

x₁₂ = -b ± $\sqrt{D}$ / 2a = -3 ± $\sqrt{441}$ / 2 * 1 = -3 ± 21 / 2

x₁ = -3 + 21 / 2 = 18 / 2 = 9

x₂ = -3 - 21 / 2 =  -24 / 2= -12

x₁ = 9

x₂ = -12

Answer 2

Квадратное уравнение №1.

Решение:

$14{x}^{2}-5x-19=0 \\ \\ D={b}^{2}-4ac={(-5)}^{2}-4\cdot14\cdot(-19)=25+1064=1089 \\ \\ {x}_{1}=\dfrac{-b-\sqrt{D}}{2a}=\dfrac{-(-5)-\sqrt{1089}}{2\cdot14}=\dfrac{5-33}{28}=\dfrac{-28}{28}=-1 \\ \\ {x}_{2}=\dfrac{-b+\sqrt{D}}{2a}=\dfrac{-(-5)+\sqrt{1089}}{2\cdot14}=\dfrac{5+33}{14}=\dfrac{38}{28}=\dfrac{19}{14}=1\dfrac{5}{14}$

Ответ: $\boxed{x_1=-1; \: \: x_2=1\dfrac{5}{14}}$

Квадратное уравнение №2.

Решение:

$y^2+3y-108=0 \\ \\ D=b^2-4ac=3^2-4\cdot1\cdot(-108)=9+432 = 441 \\ \\ x_1=\dfrac{-b-\sqrt{D}}{2a}=\dfrac{-3-\sqrt{441}}{2\cdot1}=\dfrac{-3-21}{2}=\dfrac{-24}{2}=-12 \\ \\ x_2=\dfrac{-b+\sqrt{D}}{2a}=\dfrac{-3+\sqrt{441}}{2\cdot1}=\dfrac{-3+21}{2}=\dfrac{18}{2}=9$

Ответ: $\boxed{x_1=-12; \: \: x_2=9}$