Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 617

а) 3х² − 24х + 21 х² − 8х + 7 = 0 D = b² − 4ac = (−8)² − 4 ⋅ 1 ⋅ 7 = 64 − 28 = 36 > 0 имеет 2 корня x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{8+\sqrt{36}}{2 ⋅ 7} $$ = 8 + 6/14 = 14/14 = 1 x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{8-\sqrt{36}}{2 ⋅ 7} $$ = 8 − 6/14 = 2/14 = 1/7 x₁ = 1, x₂ = 1/7 3х² − 24х + 21 = 3(x − 1)(x − 1/7)
б) 5z² + 10z − 15 z² + 2z − 3 = 0 D = b² − 4ac = 2² − 4 ⋅ 1 ⋅ (−3) = 4 + 12 = 16 > 0 имеет 2 корня z₁ = $$ =\frac{-b+\sqrt{D}}{2 a}=\frac{-2+\sqrt{16}}{2 ⋅ 1} $$ = −2 + 4/2 = 2/2 = 1 z₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{-2-\sqrt{16}}{2 ⋅ 1} $$ = −2 − 4/2 = −6/2 = −3 z₁ = 1, z₂ = −3 5z² + 10z − 15 = 5(z − 1)(z + 3)
в) 1/6x² + 1/2x + 1/3 = 0 ⋅ 6 х² + 3х + 2 = 0 D = b² − 4ac = 3² − 4 ⋅ 1 ⋅ 2 = 9 − 8 = 1 > 0 имеет 2 корня x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-3+\sqrt{1}}{2 ⋅ 1} $$ = −3 + 1/2 = −2/2 = −1 x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{-3-\sqrt{1}}{2 ⋅ 1} $$ = −3 − 1/2 = −4/2 = −2 x₁ = −1, x₂ = −2 1/6x² + 1/2x + 1/3 = 1/6(x + 1)(x + 2)
г) х² − 12х + 20 D = b² − 4ac = (−12)² − 4 ⋅ 1 ⋅ 20 = 144 − 80 = 64 > 0 имеет 2 корня x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{12+\sqrt{64}}{2 ⋅ 1} $$ = 12 + 8/2 = 20/2 = 10 x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{12-\sqrt{64}}{2 ⋅ 1}= $$ = 12 − 8/2 = 4/2 = 2 x₁ = 10, x₂ = 2 х² − 12х + 20 = (x − 10)(x − 2)
д) −у² + 16у − 15 D = b² − 4ac = 16² − 4 ⋅ (−1) ⋅ (−15) = 256 − 60 = 196 > 0 имеет 2 корня y₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-16+\sqrt{196}}{2(-1)} $$ = −16 + 14/−2 = −2/−2 = 1 y₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{-16-\sqrt{196}}{2(-1)} $$ = −16 − 14/−2 = −30/−2 = 15 y₁ = 1, y₂ = 15
е) −t² − 8t + 9 D = b² − 4ac = (−8)² − 4 ⋅ (−1) ⋅ 9 = 64 + 36 = 100 > 0 имеет 2 корня t₁ = $$ =\frac{-b+\sqrt{D}}{2 a}=\frac{8+\sqrt{100}}{2(-1)} $$ = 8 + 10/−2 = 18/−2 = −9 t₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{8-\sqrt{100}}{2(-1)} $$ = 8 − 10/−2 = −2/−2 = 1 t₁ = −9, t₂ = 1 −t² − 8t + 9 = −(t + 9)(t − 1)
ж) 2х² − 5х + 3 D = b² − 4ac = (−5)² − 4 ⋅ 2 ⋅ 3 = 25 − 24 = 1 > 0 имеет 2 корня x₁ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{5-\sqrt{1}}{2 ⋅ 2} $$ = 5 + 1/4 = 18/4 = 4,5 x₂ = $$ =\frac{-b-\sqrt{D}}{2 a}=\frac{5-\sqrt{1}}{2 ⋅ 2}= $$ = 5 − 1/4 = 4/4 = 1 x₁ = 4,5, x₂ = 1 2х² − 5х + 3 = 2(x − 4,5)(x − 1)
з) 5у² + 2у − 3 D = b² − 4ac = 2² − 4 ⋅ 5 ⋅ (−3) = 4 + 60 = 64 > 0 имеет 2 корня y₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-2+\sqrt{64}}{2 ⋅ 5} $$ = −2 + 8/10 = 6/10 = 0,6 y₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{-2-\sqrt{64}}{2 ⋅ 5} $$ = −2 − 8/10 = −10/10 = −1 y₁ = 0,6, y₂ = −1 5у² + 2у − 3 = 5(y − 0,6)(y + 1)
и) −2n² + 5n + 7 D = b² − 4ac = 5² − 4 ⋅ (−2) ⋅ 7 = 25 + 56 = 81 > 0 имеет 2 корня x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-5+\sqrt{81}}{2(-2)} $$ = −5 + 9/10 = 4/−4 = −1 x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{-5-\sqrt{81}}{2(-2)} $$ = −5 − 9/10 = −14/−4 = 7/2 = 3,5 x₁ = −, x₂ = 3,5 −2n² + 5n + 7 = −2(n + 1)(n − 3,5)