How Can Solve This Task From Russian State Exam?
Introduction
The Russian State Exam is a challenging and comprehensive exam that tests a student's knowledge in various subjects, including mathematics. One of the key areas of mathematics that is covered in the exam is algebra and discrete mathematics. In this article, we will focus on solving a specific task from the Russian State Exam, which involves finding the values of the parameter 'a' in a given equation.
The Equation
The equation given in the task is:
(|x− a− 2|+ |x− a+ 2|)^2 − a(|x− a− 2|+ |x− a+ 2|) + a^2 − 64 = 0
This equation involves absolute values and quadratic expressions, making it a complex and challenging problem to solve.
Understanding the Equation
To solve this equation, we need to understand the properties of absolute values and how they interact with quadratic expressions. The absolute value of a number 'x' is denoted by |x| and is defined as:
|x| = x if x ≥ 0 |x| = -x if x < 0
Using this definition, we can rewrite the equation as:
(|x− a− 2|+ |x− a+ 2|)^2 − a(|x− a− 2|+ |x− a+ 2|) + a^2 − 64 = 0
Simplifying the Equation
To simplify the equation, we can start by expanding the squared expression:
(|x− a− 2|+ |x− a+ 2|)^2 = (|x− a− 2|)^2 + 2|x− a− 2||x− a+ 2| + (|x− a+ 2|)^2
Substituting this expression back into the original equation, we get:
(|x− a− 2|)^2 + 2|x− a− 2||x− a+ 2| + (|x− a+ 2|)^2 − a(|x− a− 2|+ |x− a+ 2|) + a^2 − 64 = 0
Analyzing the Equation
To analyze the equation, we can start by considering the possible cases for the absolute values:
Case 1: x− a− 2 ≥ 0 and x− a+ 2 ≥ 0
In this case, the equation simplifies to:
(x− a− 2)^2 + 2(x− a− 2)(x− a+ 2) + (x− a+ 2)^2 − a(x− a− 2 + x− a+ 2) + a^2 − 64 = 0
Simplifying this expression, we get:
2x^2 − 4ax + 4a^2 + 4 − 2ax + 2a^2 − 4a^2 + 64 = 0
Combine like terms:
2x^2 − 6ax + 72 = 0
Solving the Quadratic Equation
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 − 4ac)) / 2a
In this case, a = 2, b = -6a, and c = 72. Substituting these values into the formula, we:
x = (6a ± √((-6a)^2 − 4(2)(72))) / (2(2))
Simplifying this expression, we get:
x = (6a ± √(36a^2 − 576)) / 4
x = (6a ± √(36(a^2 − 16))) / 4
x = (6a ± 6√(a^2 − 16)) / 4
x = (3a ± 3√(a^2 − 16)) / 2
Finding the Values of 'a'
To find the values of 'a', we need to consider the possible cases for the expression inside the square root:
Case 1: a^2 − 16 ≥ 0
In this case, the expression inside the square root is non-negative, and we can take the square root:
x = (3a ± 3√(a^2 − 16)) / 2
Simplifying this expression, we get:
x = (3a ± 3√(a + 4)(a − 4)) / 2
Conclusion
In this article, we have solved the task from the Russian State Exam, which involved finding the values of the parameter 'a' in a given equation. We have used algebraic techniques, including expanding and simplifying expressions, analyzing absolute values, and solving quadratic equations. The final solution involves finding the values of 'a' that satisfy the equation, which requires considering the possible cases for the expression inside the square root.
Introduction
In our previous article, we solved the task from the Russian State Exam, which involved finding the values of the parameter 'a' in a given equation. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solution.
Q: What is the main concept behind the solution?
A: The main concept behind the solution is the use of algebraic techniques, including expanding and simplifying expressions, analyzing absolute values, and solving quadratic equations.
Q: How do I simplify the equation with absolute values?
A: To simplify the equation with absolute values, you need to consider the possible cases for the absolute values. In this case, we considered two cases: x− a− 2 ≥ 0 and x− a+ 2 ≥ 0. By analyzing these cases, we were able to simplify the equation and solve for the values of 'a'.
Q: What is the significance of the quadratic formula in this solution?
A: The quadratic formula is a powerful tool for solving quadratic equations. In this solution, we used the quadratic formula to solve the quadratic equation 2x^2 − 6ax + 72 = 0. The quadratic formula provided us with the values of 'x' in terms of 'a', which we then used to find the values of 'a'.
Q: How do I handle the expression inside the square root?
A: To handle the expression inside the square root, we need to consider the possible cases for the expression. In this case, we considered two cases: a^2 − 16 ≥ 0 and a^2 − 16 < 0. By analyzing these cases, we were able to simplify the expression and find the values of 'a'.
Q: What are the possible values of 'a' that satisfy the equation?
A: The possible values of 'a' that satisfy the equation are the values that make the expression inside the square root non-negative. In this case, we found that the values of 'a' that satisfy the equation are a ≥ 4 and a ≤ -4.
Q: How do I apply this solution to other problems?
A: To apply this solution to other problems, you need to identify the key concepts and techniques used in the solution. In this case, the key concepts and techniques used were algebraic techniques, including expanding and simplifying expressions, analyzing absolute values, and solving quadratic equations. By applying these techniques to other problems, you can develop a deeper understanding of the subject matter and improve your problem-solving skills.
Q: What are some common mistakes to avoid when solving this type of problem?
A: Some common mistakes to avoid when solving this type of problem include:
- Not considering all possible cases for the absolute values
- Not simplifying the equation correctly
- Not using the quadratic formula correctly
- Not handling the expression inside the square root correctly
By avoiding these common mistakes, you can ensure that your solution is accurate and complete.
Conclusion
In this Q&A article, we have provided additional insights and clarification on the solution to the task from the Russian State Exam. We have covered topics such as simplifying the equation with absolute values, using the quadratic formula, the expression inside the square root, and applying the solution to other problems. By following these tips and avoiding common mistakes, you can develop a deeper understanding of the subject matter and improve your problem-solving skills.