Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 702

а) $$ \left\{\begin{array} { l } { y ^ { 2 } - x = - 1 } \\ { x = y + 3 } \end{array} \left\{\begin{array}{l} y^2-(y+3)=-1 \\ x=y+3 \end{array}\right.\right. $$ $$ \left\{\begin{array} { l } { y ^ { 2 } - y - 3 + 1 = 0 } \\ { x = y + 3 } \end{array} \left\{\begin{array}{l} y^2-y-2=0 \\ x=y+3 \end{array}\right.\right. $$
у² − у − 2 = 0 D = b² − 4ac = (−1)² − 4 ⋅ 1 ⋅ (−2) = 1 + 8 = 9 > 0, имеет 2 корня
y₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{1+\sqrt{9}}{2 ⋅ 1} $$ = 1 + 3/2 = 4/2 = 2 y₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{1-\sqrt{9}}{2 ⋅ 1} $$ = 1 − 3/2 = −2/2 = −1
y₁ = 2, y₂ = −1
$$ \left\{\begin{array} { l } { y _ { 1 } = 2 } \\ { x _ { 1 } = 2 + 3 = 5 } \end{array} \left\{\begin{array}{l} y_1=2 \\ x_1=5 \end{array}\right.\right. $$
$$ \left\{\begin{array} { l } { y _ { 2 } = - 1 } \\ { x _ { 2 } = - 1 + 3 = 5 } \end{array} \left\{\begin{array}{l} y_2=-1 \\ x_2=2 \end{array}\right.\right. $$
(5; 2), (2; −1)
б) $$ \left\{\begin{array} { l } { y = x - 1 } \\ { x ^ { 2 } - 2 y = 2 6 } \end{array} \left\{\begin{array}{l} y=x-1 \\ x^2-2(x-1)=26 \end{array}\right.\right. $$ $$ \left\{\begin{array} { l } { y = x - 1 } \\ { x ^ { 2 } - 2 x + 2 - 2 6 = 0 } \end{array} \left\{\begin{array}{l} y=x-1 \\ x^2-2 x-24=0 \end{array}\right.\right. $$
х² − 2х − 24 = 0 D = b² − 4ac = (−2)² − 4 ⋅ 1 ⋅ (−24) = 4 + 96 = 100 > 0, имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{2+\sqrt{100}}{2 ⋅ 1} $$ = 2 + 10/2 = 12/2 = 6 x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{2-\sqrt{100}}{2 ⋅ 1} $$ = 2 − 10/2 = −8/2 = −4
x₁ = 6, x₂ = −4
$$ \left\{\begin{array} { l } { x _ { 1 } = 6 } \\ { y _ { 1 } = 6 - 1 = 5 } \end{array} \left\{\begin{array}{l} x_1=6 \\ y_1=5 \end{array}\right.\right. $$
$$ \left\{\begin{array} { l } { x _ { 2 } = - 4 } \\ { y _ { 2 } = - 4 - 1 = - 5 } \end{array} \left\{\begin{array}{l} x_2=-4 \\ y_2=-5 \end{array}\right.\right. $$
(6; 5), (−4; −5)
в) $$ \left\{\begin{array} { l } { x y + x = - 4 } \\ { x - y = 6 } \end{array} \left\{\begin{array}{l} x(x-6)+x=-4 \\ y=x-6 \end{array}\right.\right. $$ $$ \left\{\begin{array} { l } { x ^ { 2 } - 6 x + x + 4 = 0 } \\ { y = x - 6 } \end{array} \left\{\begin{array}{l} x^2-5 x+4=0 \\ y=x-6 \end{array}\right.\right. $$
х² − 5х + 4 = 0 D = b² − 4ac = (−5)² − 4 ⋅ 1 ⋅ 4 = 25 − 16 = 9 > 0, имеет 2 корня
x₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{5+\sqrt{9}}{2 ⋅ 1} $$ = 5 + 3/2 = 8/2 = 4 x₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{5-\sqrt{9}}{2 ⋅ 1} $$ = 5 − 3/2 = 2/2 = 1
x₁ = 4, x₂ = 1
$$ \left\{\begin{array} { l } { x _ { 1 } = 4 } \\ { y _ { 1 } = 4 - 6 = - 2 } \end{array} \left\{\begin{array}{l} x_1=4 \\ y_1=-2 \end{array}\right.\right. $$
$$ \left\{\begin{array} { l } { x _ { 2 } = 1 } \\ { y _ { 2 } = 1 - 6 = - 5 } \end{array} \left\{\begin{array}{l} x_2=1 \\ y_2=-5 \end{array}\right.\right. $$
(4; −2), (1; −5)
г) $$ \left\{\begin{array} { l } { x + y = 9 } \\ { y ^ { 2 } + x = 2 9 } \end{array} \left\{\begin{array}{l} x=9-y \\ y^2+x=29 \end{array}\right.\right. $$ $$ \left\{\begin{array} { l } { x = 9 - y } \\ { y ^ { 2 } + 9 - y - 2 9 = 0 } \end{array} \left\{\begin{array}{l} x=9-y \\ y^2-y-20=0 \end{array}\right.\right. $$
у² − у − 20 = 0 D = b² − 4ac = (−1)² − 4 ⋅ 1 ⋅ (−20) = 1 + 80 = 81 > 0, имеет 2 корня
y₁ = $$ \frac{-b+\sqrt{D}}{2 a}=\frac{1+\sqrt{81}}{2 ⋅ 1} $$ = 1 + 9/2 = 10/2 = 5 y₂ = $$ \frac{-b-\sqrt{D}}{2 a}=\frac{1-\sqrt{81}}{2 ⋅ 1} $$ = 1 − 9/2 = −8/2 = −4
y₁ = 5, y₂ = −4
$$ \left\{\begin{array} { l } { y _ { 1 } = 5 } \\ { x _ { 1 } = 9 - 5 = 4 } \end{array} \left\{\begin{array}{l} y_1=5 \\ x_1=4 \end{array}\right.\right. $$
$$ \left\{\begin{array} { l } { y _ { 2 } = - 4 } \\ { x _ { 2 } = 9 + 4 = 1 3 } \end{array} \left\{\begin{array}{l} y_2=-4 \\ x_2=13 \end{array}\right.\right. $$
(4; 5), (13; −4)