Question

При каких значениях а уравнение ах-2а-2=0 имеет корень меньший 1?

Answer 1

ах-2а-2=0выразим х

ax=2a+2

x=(2a+2)/a

(2a+2)/a<1

((2a+2)-a)/a<0

(a+2)/a<0

a не равно нулю, так как на 0 делить нельзя

а+2=0

а=-2

наносим -2 и 0 на числовую прямую, и смотрим знак методом пробной точки

выбираем где <0

ответ а принадлежит (-2 ,0)

Answer 2

Equation Solution for \(ax-2a-2=0\)

To find the values of \(a\) for which the equation \(ax-2a-2=0\) has a root less than 1, we can use the quadratic formula. The quadratic formula states that for an equation in the form \(ax^2 + bx + c = 0\), the solutions are given by:

\[ x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \]

In this case, the equation is \(ax-2a-2=0\), so we can identify \(a = a\), \(b = -2a\), and \(c = -2\). We want to find the values of \(a\) for which the solutions are less than 1.

Using the Quadratic Formula

Applying the quadratic formula to the given equation, we get:

\[ x = \frac{{-(-2a) \pm \sqrt{{(-2a)^2-4a(-2)}}}}{{2a}} \]

Simplifying further:

\[ x = \frac{{2a \pm \sqrt{{4a^2+8a}}}}{{2a}} \]

This simplifies to:

\[ x = \frac{{2a \pm 2\sqrt{{a^2+2a}}}}{{2a}} \]

Simplifying the expression, we get:

\[ x = 1 \pm \sqrt{{1+\frac{2}{a}}} \]

Conditions for Roots Less Than 1

For the roots to be less than 1, the following conditions must be met:

1. \(1 + \sqrt{{1+\frac{2}{a}}} < 1\) 2. \(1 - \sqrt{{1+\frac{2}{a}}} < 1\)

Solving these inequalities will give us the range of values for \(a\) for which the equation has a root less than 1.

Conclusion

By solving the inequalities derived from the quadratic formula, we can determine the values of \(a\) for which the equation \(ax-2a-2=0\) has a root less than 1.