Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 543

а) (x + 4)² = 3x + 40 x² + 8x + 16 = 3x + 40 x² + 8x + 16 − 3x − 40 = 0 x² + 5x − 24 = 0 D = b² − 4ac = 5² − 4 ⋅ 1 ⋅ (−24) = 25 +96 = 121 > 0, уравнение имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-5+\sqrt{121}}{2 ⋅ 1} $$ = −5 + 11/2 = 6/2 = 3
x₂ = $$ =\frac{-b+\sqrt{D}}{2 a}=\frac{-5-\sqrt{121}}{2 ⋅ 1} $$ = −5 − 11/2 = −16/2 = −8
x₁ = 3, x₂ = −8
б) (2p − 3)² = 11p − 19 4p² − 12p + 9 = 11p − 19 4p² − 12p + 9 − 11p + 19 = 0 4p² − 23p + 28 = 0 D = b² − 4ac = (−23)² − 4 ⋅ 4 ⋅ 28 = 529 − 448 = 81 > 0, уравнение имеет 2 корня
p₁ = $$ =\frac{-b+\sqrt{D}}{2 a}=\frac{23+\sqrt{81}}{2 ⋅ 4} $$ = 23 + 9/8 = 32/8 = 4
p₂ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{23-\sqrt{81}}{2 ⋅ 4} $$ = 23 − 9/8 = 12/8 = 1,5
p₁ = 4, p₂ = 1,5
в) 3(x + 4)² = 10x + 32 3(x² + 8x + 16) = 10x + 32 3x² + 24x + 48 − 10x − 32 = 0 3x² + 14x + 16 = 0 D = b² − 4ac = 14² − 4 ⋅ 3 ⋅ 16 = 196 -192 = 4 > 0, уравнение имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-14+\sqrt{4}}{2 ⋅ 3}= $$ = −14 + 2/6 = 12/6 = 2
x₂ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-14-\sqrt{4}}{2 ⋅ 3} $$ = −14 − 2/6 = −16/6 = −22/3
x₁ = 2, x₂ = −22/3
г) 15y² + 17 = 15(y + 1)² 15y² + 17 = 15(y² + 2y + 1) 15y² + 17 = 15y² + 30y + 15 15y² + 17 − 15y² − 30y − 15 = 0 2 − 30y = 0 30y = 2 y = 2 : 30 y = 2/30 = 1/15
д) (x + 1)² = 7918 − 2x x² + 2x + 1 = 7918 − 2x x² + 2x + 1 − 7918 + 2x = 0 x² + 4x − 7917 = 0 D = b² − 4ac = 4² − 4 ⋅ 1 ⋅ (−7917) = 16 + 31668 = 31684 > 0, уравнение имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-4+\sqrt{31684}}{2 ⋅ 1} $$ = −4 + 178/2 = 174/2 = 87
x₂ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-4-\sqrt{31684}}{2 ⋅ 1} $$ = −4 − 178/2 = −182/2 = −91
x₁ = 87, x₂ = −91
е) (m + 2)² = 3131 − 2m m² + 4m + 4 = 3131 − 2m m² + 4m + 4 − 3131 + 2m = 0 m² + 6m − 3127 = 0 D = b² − 4ac = 6² − 4 ⋅ 1 ⋅ (−3127) = 36 + 12508 = 12544 > 0, уравнение имеет 2 корня
m₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-6+\sqrt{12544}}{2 ⋅ 1} $$ = −6 + 112/2 = 106/2 = 53
m₂ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{-6-\sqrt{12544}}{2 ⋅ 1} $$ = −6 − 112/2 = −118/2 = −59
m₁ = 53, m₂ = −59
ж) (x + 1)² = (2x − 1)² x² + 2x + 1 = 4x² − 4x + 1 x² + 2x + 1 − 4x² + 4x − 1 = 0 −3x² + 6x = 0 −3x(x + 2) = 0 x₁ = 0 x + 2 = 0 x₂ = −2
з) (n − 2)² + 48 = (2 − 3n)² n² − 4n + 4 + 48 = 4 − 12n + 9n² n² − 4n + 4 + 48 − 4 + 12n − 9n² = 0 −8n² + 8n + 48 = 0 : (−8) n² − n − 6 = 0 D = b² − 4ac = (−1)² − 4 ⋅ 1 ⋅ (−6) = 1 + 24 = 25 > 0, уравнение имеет 2 корня
n₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{1+\sqrt{25}}{2 ⋅ 1} $$ = 1 + 5/2 = 6/2 = 3
n₂ = $$ =\frac{-b+\sqrt{D}}{2 a}=\frac{1-\sqrt{25}}{2 ⋅ 1} $$ = 1 − 5/2 = −4/2 = −2
n₁ = 3, n₂ = −2