Solving (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, X + 4] A Detailed Solution
Introduction
In this article, we will delve into solving the mathematical equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, x + 4]. This equation presents a unique challenge as it combines a rational function with a vector representation on the right-hand side. Understanding the nature of each component and how they interact is crucial to finding the solution. We will explore the properties of quadratic expressions, discuss the implications of vector equality, and systematically work through the steps required to solve for x. Our approach will involve algebraic manipulation, careful consideration of potential extraneous solutions, and a thorough analysis of the results. This exploration will not only provide a solution to the given equation but also enhance our understanding of mathematical problem-solving techniques applicable to a wide range of similar problems. The equation at hand serves as an excellent example of how different mathematical concepts can converge, requiring a comprehensive understanding to unravel its solution. Before we dive into the detailed steps, it’s important to recognize the key elements: the quadratic expressions in the numerator and denominator of the rational function, and the vector on the right-hand side. Each of these components brings its own set of properties and considerations that will influence our solution strategy. Let's begin by examining the rational function in more detail.
Understanding the Rational Function
To effectively tackle the equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, x + 4], it is crucial to first understand the behavior of the rational function on the left-hand side. The rational function is composed of two quadratic expressions: the numerator x^2 - 6x + 10 and the denominator x^2 + 8x + 17. Analyzing these quadratic expressions is essential for determining the domain of the function and understanding its potential values. A key aspect to consider is whether the denominator can ever be equal to zero, as this would make the function undefined. The denominator x^2 + 8x + 17 is a quadratic expression, and we can determine its roots by using the discriminant. The discriminant, given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0, provides insights into the nature of the roots. If the discriminant is negative, the quadratic expression has no real roots, indicating that the denominator will never be zero for any real value of x. For the denominator x^2 + 8x + 17, the discriminant is 8^2 - 4(1)(17) = 64 - 68 = -4, which is negative. This confirms that the denominator has no real roots, and therefore, the rational function is defined for all real values of x. Next, we can analyze the numerator x^2 - 6x + 10. Its discriminant is (-6)^2 - 4(1)(10) = 36 - 40 = -4, which is also negative. This means the numerator also has no real roots. Furthermore, since the leading coefficient of both quadratic expressions is positive, both the numerator and denominator represent upward-opening parabolas. This information is valuable as we proceed to solve the equation, as it helps us understand the possible range of values the rational function can take. Understanding these properties of the rational function sets the stage for the next step: examining the vector on the right-hand side of the equation.
Analyzing the Vector Representation
The equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, x + 4] features a vector representation on the right-hand side, which introduces a unique aspect to the problem. The expression [x - 3, x + 4] represents a vector in a two-dimensional space, where x - 3 and x + 4 are the components of the vector. However, the left-hand side of the equation, the rational function, yields a scalar value. This creates a fundamental mismatch in the equation's structure, as a scalar cannot be directly equal to a vector. This observation is crucial because it suggests that the original equation, as presented, might not have a direct solution in the traditional sense. Vector equality requires that corresponding components of two vectors be equal. In this context, the right-hand side represents a vector, while the left-hand side represents a scalar. For a scalar to be equal to a vector, the vector must be a scalar multiple of a unit vector along one of the axes, and the other component must be zero. However, in this case, we have a single scalar value equated to a two-component vector, making a direct comparison impossible. This means there is a likely error in how the equation is set up or interpreted. It's possible that the intention was to compare the scalar value from the rational function to a specific component of the vector, or that there was a misunderstanding in the notation used. Given this discrepancy, we must carefully reconsider the problem statement and look for possible interpretations or simplifications that would allow us to proceed with a solution. One way to approach this could be to assume that the equation was meant to be interpreted in a component-wise manner, where the scalar value from the rational function is somehow related to each component of the vector individually. However, without further clarification, this is just an assumption. The next step is to explore potential interpretations and approaches to resolve this issue and make the equation solvable.
Interpreting the Equation and Possible Solutions
Given the incongruity between the scalar rational function and the vector in the equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, x + 4], we need to explore potential interpretations that might lead to a meaningful solution. The most straightforward approach is to recognize that the equation, as written, is likely incorrect or incomplete. A scalar value cannot be directly equated to a vector. However, we can consider scenarios where the equation might make sense under certain assumptions or modifications. One plausible interpretation is that the equation was intended to represent a system of equations. In this scenario, the scalar value from the rational function might be related to each component of the vector separately. This would imply two separate equations:
- (x^2 - 6x + 10) / (x^2 + 8x + 17) = x - 3
- (x^2 - 6x + 10) / (x^2 + 8x + 17) = x + 4
If we consider these two equations, we can solve them independently and see if there are any common solutions for x. Solving these equations involves algebraic manipulation, which we will delve into in the next section. Another interpretation could be that the vector notation was intended to represent a range or a set of values. In this case, the equation might be asking for the values of x for which the rational function falls within the range defined by x - 3 and x + 4. However, this interpretation is less likely given the standard mathematical notation. It's also possible that the equation was meant to be a different type of relationship altogether, such as a projection or a magnitude comparison. But without additional context, these interpretations are speculative. Therefore, the most reasonable approach is to proceed with the assumption that the equation represents a system of two equations, as outlined above. This allows us to apply algebraic techniques to solve for x and determine if any valid solutions exist. We will now focus on solving the first equation in this hypothetical system.
Solving the Equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = x - 3
Let's address the first equation derived from our interpretation: (x^2 - 6x + 10) / (x^2 + 8x + 17) = x - 3. To solve this equation, we will first eliminate the rational function by multiplying both sides by the denominator, x^2 + 8x + 17. This gives us:
x^2 - 6x + 10 = (x - 3)(x^2 + 8x + 17)
Next, we expand the right side of the equation:
x^2 - 6x + 10 = x^3 + 8x^2 + 17x - 3x^2 - 24x - 51
Now, we simplify and rearrange the equation to set it equal to zero:
0 = x^3 + 8x^2 - 3x^2 - x^2 + 17x - 24x + 6x - 51 - 10
Combining like terms, we get:
0 = x^3 + 4x^2 - x - 61
This is a cubic equation, which can be challenging to solve analytically. We can attempt to find rational roots using the Rational Root Theorem. This theorem states that any rational root of the polynomial must be a factor of the constant term (-61) divided by a factor of the leading coefficient (1). Since 61 is a prime number, the possible rational roots are ±1 and ±61. We can test these values by substituting them into the cubic equation. By testing these values, we find that none of them are roots of the equation. This suggests that the roots of the cubic equation are either irrational or complex. Solving cubic equations generally involves more advanced techniques, such as Cardano's method or numerical methods. However, for the sake of this discussion, we will acknowledge that finding the exact solutions to this cubic equation is beyond the scope of a simple analytical solution. We would need to employ numerical methods or computational tools to approximate the roots. Now, let's move on to the second equation in our system and see if it yields any simpler solutions.
Solving the Equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = x + 4
Now, let's consider the second equation derived from our initial interpretation: (x^2 - 6x + 10) / (x^2 + 8x + 17) = x + 4. Similar to the previous equation, we'll begin by multiplying both sides by the denominator, x^2 + 8x + 17, to eliminate the rational function:
x^2 - 6x + 10 = (x + 4)(x^2 + 8x + 17)
Next, we expand the right side of the equation:
x^2 - 6x + 10 = x^3 + 8x^2 + 17x + 4x^2 + 32x + 68
Now, we simplify and rearrange the equation to set it equal to zero:
0 = x^3 + 8x^2 + 4x^2 - x^2 + 17x + 32x + 6x + 68 - 10
Combining like terms, we get:
0 = x^3 + 11x^2 + 55x + 58
This is another cubic equation. Again, we can attempt to find rational roots using the Rational Root Theorem. The possible rational roots are the factors of 58 (±1, ±2, ±29, ±58). By testing these values, we find that x = -2 is a root of the equation:
(-2)^3 + 11(-2)^2 + 55(-2) + 58 = -8 + 44 - 110 + 58 = 0
Since x = -2 is a root, we can perform polynomial division to factor the cubic equation. Dividing x^3 + 11x^2 + 55x + 58 by (x + 2), we get x^2 + 9x + 29. Now we have:
0 = (x + 2)(x^2 + 9x + 29)
To find the remaining roots, we solve the quadratic equation x^2 + 9x + 29 = 0 using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-9 ± √(9^2 - 4(1)(29))) / 2(1)
x = (-9 ± √(81 - 116)) / 2
x = (-9 ± √(-35)) / 2
The roots of the quadratic equation are complex: x = (-9 ± i√35) / 2. Thus, the solutions to the second equation are x = -2 and x = (-9 ± i√35) / 2. Now, we need to check if any of these solutions also satisfy the first cubic equation we derived earlier. If a solution satisfies both equations, it would be a valid solution to the original problem, under our interpretation of the equation as a system of two equations. Let's analyze the solutions we've found.
Comparing Solutions and Final Answer
We have solved two cubic equations arising from our interpretation of the original equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, x + 4] as a system of two equations. The first equation, (x^2 - 6x + 10) / (x^2 + 8x + 17) = x - 3, led to the cubic equation x^3 + 4x^2 - x - 61 = 0. We found that this equation has no rational roots, and solving it would require numerical methods or more advanced techniques. The second equation, (x^2 - 6x + 10) / (x^2 + 8x + 17) = x + 4, led to the cubic equation x^3 + 11x^2 + 55x + 58 = 0. We found one real solution, x = -2, and two complex solutions, x = (-9 ± i√35) / 2. To determine if there is a common solution to both equations, we need to check if x = -2 is a solution to the first cubic equation. Substituting x = -2 into x^3 + 4x^2 - x - 61, we get:
(-2)^3 + 4(-2)^2 - (-2) - 61 = -8 + 16 + 2 - 61 = -51
Since the result is not zero, x = -2 is not a solution to the first cubic equation. This means that there is no common solution between the two equations. Given our interpretation of the original equation as a system of two equations, this implies that there is no solution to the original problem under this interpretation. It is important to reiterate that the original equation, equating a scalar to a vector, is likely ill-posed. Our attempt to interpret it as a system of equations was an effort to find a meaningful solution, but it ultimately did not yield a valid result. In conclusion, based on our analysis and the likely misinterpretation of the original equation, we can state that there is no solution to the equation (x^2 - 6x + 10) / (x^2 + 8x + 17) = [x - 3, x + 4] as it is presented. The incongruity between the scalar and vector representations makes a direct solution impossible, and our attempt to interpret it as a system of equations did not yield a common solution.
