Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 580

а) х² − 15х − 16 = 0 D = b² − 4ac = (−15)² − 4 ⋅ 1 ⋅ (−16) = 225 + 64 = 289 > 0, уравнение имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{15+\sqrt{289}}{2 ⋅ 1} $$ = 15 + 17/2 = 32/2 = 16
x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{15-\sqrt{289}}{2 ⋅ 1} $$ = 15 − 17/2 = −2/2 = −1
По обратной теореме Виета x₁ + x₂ = −−15/1 = 15 x₁ + x₂ = 16 + (−1) = 15, верно
x₁x₂ = −16/1 = −16 x₁x₂ = 16 ⋅ (−1) = −16, верно
б) m² − 6m − 11 = 0 D = b² − 4ac = (−6)² − 4 ⋅ 1 ⋅ (−11) = 36 + 44 = 80 > 0, уравнение имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{6+\sqrt{80}}{2 ⋅ 1}=\frac{6+4 \sqrt{5}}{2} $$ = 3 + 2$$\sqrt{5}$$
x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{6-\sqrt{80}}{2 ⋅ 1}=\frac{6-4 \sqrt{5}}{2} $$ = 3 − 2$$\sqrt{5}$$
По обратной теореме Виета x₁ + x₂ = −−6/1 = 6 x₁ + x₂ = 3 + 2$$\sqrt{5}$$ + 3 − 2$$\sqrt{5}$$ = 6, верно
x₁x₂ = −11/1 = −11 x₁x₂ = (3 + 2$$\sqrt{5}$$)(3 − 2$$\sqrt{5}$$) = 9 − 2 ⋅ 10 = 9 − 20 = −11, верно
в) 12x² − 4x − 1 = 0 D = b² − 4ac = (−4)² − 4 ⋅ 12 ⋅ (−1) = 16 + 48 = 64 > 0, уравнение имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{4+\sqrt{64}}{2 ⋅ 12}= $$ = 4 + 8/24 = 12/24 = 1/2 = 0,5
x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{4-\sqrt{64}}{2 ⋅ 12} $$ = 4 − 8/24 = −4/24 = −1/6
По обратной теореме Виета x₁ + x₂ = −−4/12 = 1/3 x₁ + x₂ = 1/2 + (−1/6) = 1/2 ⋅ 3/3 − 1/6 = 3 − 1/6 = 2/6 = 1/3, верно
x₁x₂ = −1/12 = −1/12 x₁x₂ = 1/2 ⋅ 1/6 = −1/12, верно
г) t² − 6 = 0 t² = 6 t = ±$$\sqrt{6}$$ t₁ = $$\sqrt{6}$$ t₂ = −$$\sqrt{6}$$
По обратной теореме Виета t₁ + t₂ = −0/1 = 0 t₁ + t₂ = $$\sqrt{6}$$ + (−$$\sqrt{6}$$) = 0, верно
t₁t₂ = −6/1 = −6 t₁t₂ = $$\sqrt{6}$$ ⋅ (−$$\sqrt{6}$$), верно
д) 5x² − 18x = 0 х(5х − 18) = 0 х₁ = 0 5х − 18 = 0 5х = 18 х = 18 : 5 = 3,6 х₂ = 3,6
По обратной теореме Виета x₁ + x₂ = −18/5 = 3,6 x₁ + x₂ = 0 + 3,6 = 3,6, верно
x₁x₂ = 0/5 = 0 x₁x₂ = 0 ⋅ 3,6 = 0, верно
е) 2y² − 41 = 0 2y² = 41 y² = 41 : 2 = 41/2 y = ±$$ \pm \sqrt{\frac{41}{2}} $$ y₁ = $$ \pm \sqrt{\frac{41}{2}} $$ y₂ = −$$ \pm \sqrt{\frac{41}{2}} $$
По обратной теореме Виета y₁ + y₂ = −0/2 = 0 y₁ + y₂ = $$ \sqrt{\frac{41}{2}}+\left(-\sqrt{\frac{41}{2}}\right)= $$ = 0, верно
y₁y₂ = −41/2 = −20,5 y₁y₂ = $$ \pm \sqrt{\frac{41}{2}} $$ ⋅ (−$$ \pm \sqrt{\frac{41}{2}} $$) = −41/2 = −20,5, верно