Question

Будет ли число 1 корнем уравнения ax^2-(a+c)x+c=0? Каков его второй корень

Answer 1

$D=(a+c)^2-4ac=a^2+2ac+c^2-4ac=a^2-2ac+c^2=(a-c)^2 \\ \sqrt{D}= \sqrt{(a-c)^2}=|a-c| \\ x_1= \frac{a+c+|a-c|}{2a} \\ x_2= \frac{a+c-|a-c|}{2a}$
Строго говоря, для раскрытия модуля придется рассмотреть два случая.
Первый случай a≥c, тогда a-c≥0 и модуль раскрывается со знаком плюс:
$x_1= \frac{a+c+a-c}{2a} = 1 \\ x_2= \frac{a+c-a+c|}{2a} = \frac{2c}{2a} = \frac{c}{a}$

Второй случай a<c, тогда a-c<0 и модуль раскрывается со знаком минус:
$x_1= \frac{a+c-a+c}{2a} =  \frac{2c}{2a} = \frac{c}{a} \\ x_2= \frac{a+c+a-c|}{2a} = 1$
Ответ: 1; $\frac{c}{a}$

Answer 2

Будет потому что сумма коэффициентов равна 0 а-а-с+с=0
второй корень по теореме виета равен с/а

Answer 3

Will the number 1 be a root of the equation ax^2-(a+c)x+c=0? What is its second root?

To determine if the number 1 is a root of the equation ax^2-(a+c)x+c=0, we can substitute x=1 into the equation and check if it satisfies the equation.

Let's substitute x=1 into the equation:

ax^2-(a+c)x+c=0

a(1)^2-(a+c)(1)+c=0

Simplifying the equation, we get:

a-a-c+c=0

0=0

Since the equation simplifies to 0=0 when x=1, we can conclude that the number 1 is indeed a root of the equation.

Now, let's find the second root of the equation. To do this, we can use the quadratic formula:

x = (-b ± √(b^2-4ac)) / (2a)

In this case, the equation is ax^2-(a+c)x+c=0, so we have:

a = a b = -(a+c) c = c

Substituting these values into the quadratic formula, we get:

x = (-(a+c) ± √((a+c)^2-4ac)) / (2a)

Simplifying further, we have:

x = (-a-c ± √(a^2+2ac+c^2-4ac)) / (2a)

x = (-a-c ± √(a^2-2ac+c^2)) / (2a)

x = (-a-c ± √((a-c)^2)) / (2a)

x = (-a-c ± (a-c)) / (2a)

Now, let's consider the two cases:

Case 1: x = (-a-c + (a-c)) / (2a)

Simplifying, we have:

x = (-a-c + a-c) / (2a)

x = (-2c) / (2a)

x = -c / a

Case 2: x = (-a-c - (a-c)) / (2a)

Simplifying, we have:

x = (-a-c - a+c) / (2a)

x = (-2a) / (2a)

x = -1

Therefore, the second root of the equation ax^2-(a+c)x+c=0 is -c/a.

To summarize: - The number 1 is a root of the equation ax^2-(a+c)x+c=0. - The second root of the equation is -c/a.