Question

Какие числа при умножении дают -3 а при сложении 4

Answer 1

Пусть a и b - искомые числа

Тогда

$\left \{ {{a\cdot b=-3} \atop {a+ b = 4}} \right. \left \{ {{a\cdot (4-a)=-3} \atop {b = 4-a}} \right. \left \{ {{4a-a^2=-3} \atop {b = 4-a}} \right. \left \{ {{a^2-4a-3=0} \atop {b = 4-a}} \right. \\ \\ a^2-4a-3=0 \\ \\ a_{1,2}=\frac{-(-4)\pm\sqrt{(-4)^2-4\cdot 1 \cdot (-3)}}{2\cdot 1}=\frac{4\pm \sqrt{16+12}}{2}=\frac{4\pm\sqrt{28}}{2}=\frac{4\pm\sqrt{7\cdot4}}{2}=\frac{4\pm2\sqrt{7}}{2}=2\pm\sqrt{7}$

$\left \{ {{a_1=2+\sqrt{7}} \atop {b_1=4-(2+\sqrt{7})}} \right. \left \{ {{a_1=2+\sqrt{7}} \atop {b_1=4-2-\sqrt{7}}} \right. \left \{ {{a_1=2+\sqrt{7}} \atop {b_1=2-\sqrt{7}}} \right. \\ \\ \left \{ {{a_2=2-\sqrt{7}} \atop {b_2=4-(2-\sqrt{7})}} \right. \left \{ {{a_2=2-\sqrt{7}} \atop {b_2=4-2+\sqrt{7}}} \right. \left \{ {{a_2=2-\sqrt{7}} \atop {b_2=2+\sqrt{7}}} \right.$