Question

Здравствуйте. Нужна помощь. Я изучаю параметры, но у меня это очень плохо получается. Например, есть вот такой вот параметр: 7(2x-1)a^2-(23x-22)a+3(x-1)=0 Я раскрыла скобки и преобразовала это выражение, вот что получилось: x(14a^2-23a+3)=7a^2-22a+3 Далее я нашла корни двух квадратных уравнений: x(7a-1)(a-1,5)=(7a-1)(a-3) x(7a-1)(a-1,5)-(a-3)=0 А что делать дальше я не знаю. Для каждого значения параметра a найдите число корней уравнения. Почему при a=1/7 уравнение не имеет корней? Почему при a=1.5 нет корней? Как я должна дальше рассуждать в таких задачах и откуда все это берется???? Заранее спасибо за ответ.

Answer 1

У Вас есть в решении ошибки. Правильно так:

(7a-1)(2a-3)x=(7a-1)(a-3).

1 случай:  a=1/7. Уравнение превращается в 0=0, то есть подходит любой  x;   решений поэтому бесконечно много.

Теперь можно считать, что a не равен 1/7 и сократить на (7a-1); получается уравнение (2a-3)x=a-3.

2 случай: a=3/2. Уравнение превращается в 0= - 3/2, поэтому оно решений не имеет.

3 случай: a  не равен 1/7 и не равен 3/2, тогда x=(a-3)/(2a-3) - единственное решение.

Answer 2

Understanding the Problem

You are studying parameters and are having difficulty with a specific problem involving an equation. The equation is given as: 7(2x-1)a^2-(23x-22)a+3(x-1)=0. You have expanded and simplified the equation to: x(14a^2-23a+3)=7a^2-22a+3. You have also found the roots of two quadratic equations: x(7a-1)(a-1.5)=(7a-1)(a-3). However, you are unsure about what to do next and why the equation has no roots when a=1/7 and a=1.5. You are seeking guidance on how to approach such problems and where all this information comes from.

Analyzing the Equation

Let's analyze the equation step by step to understand it better.

The given equation is: 7(2x-1)a^2-(23x-22)a+3(x-1)=0.

You have expanded and simplified it to: x(14a^2-23a+3)=7a^2-22a+3.

Finding the Roots

Next, you found the roots of two quadratic equations: x(7a-1)(a-1.5)=(7a-1)(a-3).

To find the roots, you set each factor equal to zero:

1. x = 0 2. 7a - 1 = 0, which gives a = 1/7 3. a - 1.5 = 0, which gives a = 1.5

Analyzing the Roots

Now, let's analyze why the equation has no roots when a = 1/7 and a = 1.5.

When a = 1/7, the equation becomes: x(14(1/7)^2 - 23(1/7) + 3) = 7(1/7)^2 - 22(1/7) + 3.

Simplifying this further, we get: x(2 - 23/7 + 3) = 1/7 - 22/7 + 3.

Simplifying the equation, we have: x(2 - 23/7 + 3) = -20/7 + 3.

Further simplification gives: x(2 - 23/7 + 3) = -20/7 + 21/7.

Combining the terms, we have: x(2 - 23/7 + 3) = 1/7.

Simplifying the equation, we get: x(2 - 23/7 + 3) = 1/7.

Since the left side of the equation is always positive (as it is the product of x and a positive value), and the right side of the equation is positive (1/7), there are no real solutions for x that satisfy this equation. Therefore, when a = 1/7, the equation has no roots.

Similarly, when a = 1.5, the equation becomes: x(14(1.5)^2 - 23(1.5) + 3) = 7(1.5)^2 - 22(1.5) + 3.

Simplifying this further, we get: x(2.25 - 34.5 + 3) = 15.75 - 33 + 3.

Simplifying the equation, we have: x(2.25 - 34.5 + 3) = -14.25 + 3.

Further simplification gives: x(2.25 - 34.5 + 3) = -11.25.

Since the left side of the equation is always positive (as it is the product of x and a positive value), and the right side of the equation is negative (-11.25), there are no real solutions for x that satisfy this equation. Therefore, when a = 1.5, the equation has no roots.

Further Reasoning

In such problems, it is important to understand the properties of the equation and the values of the parameters involved. By analyzing the equation and the values of a, you can determine whether the equation has real solutions or not.

In this case, when a = 1/7 and a = 1.5, the equation does not have real solutions. This can be determined by analyzing the signs of the terms on both sides of the equation.

To further develop your understanding of these types of problems, it would be helpful to study quadratic equations, their properties, and how to find their roots. Additionally, practicing solving similar problems will enhance your problem-solving skills.

I hope this explanation helps! If you have any further questions, feel free to ask.