Question

Докажите что если a+b+c=0, то уравнение ax^2 +bx +c=0 , имеет корни x1=1,x2=c/a

Answer 1

$ax^2 +bx +c=0 | + (a+c)x$

$ax^2 +(b+a+c)x +c=(a+c)x\\\\ ax^2-(a+c)x +c = 0\\\\ D = (a+c)^2-4ac\\\\ x_1 = \frac{a+c+\sqrt{(a+c)^2-4ac}}{2a} = \frac{a+c+\sqrt{a^2+2ac+c^2-4ac}}{2a} =\\\\ \frac{a+c+\sqrt{a^2-2ac+c^2}}{2a} = \frac{a+c+\sqrt{(a-c)^2}}{2a} = \frac{a+c+a-c}{2a} = \frac{2a}{2a} = 1\\\\ x_2 = \frac{a+c-\sqrt{(a+c)^2-4ac}}{2a} = \frac{a+c-\sqrt{a^2+2ac+c^2-4ac}}{2a} =\\\\\frac{a+c-\sqrt{a^2-2ac+c^2}}{2a} = \frac{a+c-\sqrt{(a-c)^2}}{2a} = \frac{a+c-a+c}{2a} = \frac{2c}{2a} = \frac{c}{a}$