Why Are We Taking Discriminant Zero For T 2 + A T + 3 T^2 +at +3 T 2 + A T + 3 , Where T = X 1 + X 2 \frac{x}{1 + X^2} 1 + X 2 X
Introduction to Discriminant Analysis in Quadratic Equations
When we delve into the fascinating realm of quadratic equations, understanding the nature of their roots is paramount. At the heart of this understanding lies the discriminant, a powerful tool that unveils the secrets hidden within the equation. In this comprehensive exploration, we will dissect the significance of taking the discriminant zero for the quadratic equation t² + at + 3, where t is ingeniously defined as x / (1 + x²). Our journey will not only illuminate the mathematical principles at play but also provide a practical understanding of how discriminant analysis shapes our problem-solving approach in algebra and beyond.
The discriminant is a critical component of the quadratic formula, which is given by b² - 4ac for a quadratic equation of the form ax² + bx + c = 0. This expression, b² - 4ac, single-handedly dictates the nature and number of roots a quadratic equation possesses. When the discriminant is greater than zero, the equation boasts two distinct real roots, indicating that the parabola intersects the x-axis at two distinct points. A discriminant of zero, our focal point in this discussion, signifies that the quadratic equation has exactly one real root, implying that the parabola touches the x-axis at a single point, forming a tangent. Conversely, a negative discriminant unveils the presence of complex roots, where the parabola never crosses the x-axis.
In the specific context of t² + at + 3 = 0, we are venturing into a landscape where the variable t is itself a function of x, namely t = x / (1 + x²). This substitution introduces a layer of complexity, transforming a simple quadratic equation into a sophisticated interplay between quadratic forms and rational functions. The parameter 'a', as the coefficient of the linear term, plays a crucial role in determining the roots of the equation. By setting the discriminant to zero, we are essentially pinpointing the conditions under which the quadratic equation has precisely one solution in terms of t. This singular solution for t, however, must then be meticulously translated back into the realm of x, considering the functional relationship t = x / (1 + x²). This reverse mapping is where the true challenge lies, as the number of solutions in x may differ significantly from the number of solutions in t due to the nature of the function t(x).
The motivation behind examining the discriminant in such problems stems from the desire to control and predict the behavior of quadratic equations. In applied mathematics and physics, quadratic equations frequently arise in models describing motion, optimization problems, and many other real-world phenomena. A deep understanding of how the discriminant affects the roots is therefore not just an academic exercise but a practical necessity. For instance, in engineering, one might use the discriminant to determine the stability of a system or the conditions under which a projectile reaches a certain target. In economics, quadratic models can help predict market equilibrium, and the discriminant can inform about the uniqueness and stability of such equilibria.
In the following sections, we will systematically unpack the mathematical implications of setting the discriminant to zero for t² + at + 3 = 0, taking into account the nuanced relationship between t and x. We will explore how the values of 'a' dictate the existence and nature of the roots, and we will unravel the conditions under which the original equation, expressed in terms of x, has exactly two distinct real roots. This exploration will not only enhance our problem-solving skills but also deepen our appreciation for the interconnectedness of algebraic concepts and their practical applications.
Analyzing the Discriminant of t² + at + 3 = 0
When we encounter a quadratic equation of the form t² + at + 3 = 0, the first step in deciphering its behavior is to calculate the discriminant. As previously mentioned, the discriminant, denoted as Δ, is a pivotal expression within the quadratic formula, given by b² - 4ac. In our specific equation, a = 1, b = a (the coefficient of the t term), and c = 3. Consequently, the discriminant becomes:
Δ = a² - 4(1)(3) = a² - 12
The discriminant, a² - 12, now holds the key to understanding the roots of our quadratic equation in terms of t. To have real roots, the discriminant must be greater than or equal to zero. The condition Δ ≥ 0 leads us to:
a² - 12 ≥ 0
This inequality unveils the range of values for a that permit real solutions for t. Solving the inequality, we find that a² ≥ 12, which translates to |a| ≥ √12, or more simply, |a| ≥ 2√3. This condition implies that a must either be greater than or equal to 2√3 or less than or equal to -2√3 for the equation to have real roots. These boundary values, 2√3 and -2√3, are critical points where the nature of the roots transitions from real to complex or vice versa.
Now, let's zero in on the case where the discriminant is exactly zero, i.e., a² - 12 = 0. This scenario is particularly intriguing because it corresponds to the quadratic equation having exactly one real root. Setting the discriminant to zero gives us:
a² = 12
Which yields two specific values for a:
a = ±2√3
These values of a, 2√3 and -2√3, are not just arbitrary numbers; they represent the critical thresholds where the quadratic equation t² + at + 3 = 0 metamorphoses from having two distinct real roots to having a single repeated real root. Geometrically, this corresponds to the parabola y = t² + at + 3 just touching the t-axis at a single point, rather than intersecting it at two distinct points.
The significance of these discriminant conditions cannot be overstated. They are fundamental in determining not only the existence and number of roots but also their nature. Real roots, as opposed to complex roots, have tangible interpretations in real-world scenarios. For instance, in physics, they might represent the times at which a projectile hits the ground, while in economics, they could signify equilibrium prices in a market. Thus, understanding the discriminant is not just an algebraic exercise but a gateway to interpreting and solving real-world problems.
However, our analysis does not stop here. We must remember that t is not just a variable; it is a function of x, defined as t = x / (1 + x²). The solutions for t must therefore be carefully mapped back to x, considering the properties of this functional relationship. This reverse mapping is where the story becomes even more fascinating, as the number of solutions in x may not directly correspond to the number of solutions in t. In the next section, we will delve into the intricacies of this mapping and explore how it affects the ultimate solutions of the equation in terms of x.
The Transformation t = x / (1 + x²): Mapping Solutions from t to x
The substitution t = x / (1 + x²) is a cornerstone of this problem, and understanding its implications is crucial for accurately interpreting the solutions of the original equation. This seemingly simple transformation introduces a layer of complexity that we must carefully navigate. To fully appreciate the impact of this substitution, we need to analyze the behavior of the function t(x) = x / (1 + x²).
This function, t(x), is a rational function, and its characteristics significantly influence the number and nature of solutions we obtain in terms of x. Let's begin by examining its range. To find the range of t(x), we can rewrite the equation as a quadratic in x:
t(1 + x²) = x
tx² - x + t = 0
For x to be real, the discriminant of this quadratic equation must be non-negative. The discriminant, in this case, is 1 - 4t². Therefore, we have:
1 - 4t² ≥ 0
4t² ≤ 1
t² ≤ 1/4
This inequality reveals that the range of t(x) is -1/2 ≤ t ≤ 1/2. This is a pivotal piece of information because it tells us that only values of t within this range are permissible when we map back from the t-domain to the x-domain. Any solutions for t that fall outside this range are extraneous and do not correspond to real values of x.
Furthermore, it's insightful to note that the maximum value of t(x), 1/2, occurs when x = 1, and the minimum value, -1/2, occurs when x = -1. The function t(x) is also an odd function, meaning that t(-x) = -t(x). This symmetry about the origin has implications for the solutions we obtain; for every positive solution for x, there is a corresponding negative solution.
Now, let's consider how the values of t we obtain from solving t² + at + 3 = 0 relate to the solutions in x. For each value of t within the range -1/2 ≤ t ≤ 1/2, we need to solve the quadratic equation tx² - x + t = 0 for x. The number of real solutions for x depends on the discriminant of this new quadratic, which we already calculated as 1 - 4t². If 1 - 4t² > 0, we have two distinct real solutions for x; if 1 - 4t² = 0, we have one real solution; and if 1 - 4t² < 0, we have no real solutions.
This analysis highlights the complexity of the transformation t = x / (1 + x²). A single solution for t in the range -1/2 ≤ t ≤ 1/2 can yield zero, one, or two solutions for x, depending on the value of t. This underscores the importance of considering the range of t(x) when interpreting the solutions of the original equation.
In the context of the initial problem, where we are looking for exactly two distinct real roots, we need to ensure that the values of t we obtain, after setting the discriminant of t² + at + 3 = 0 to zero, lead to exactly two distinct real values of x. This requires a careful balancing act: the values of a must be chosen such that the discriminant of the original quadratic equation in t is zero, and the resulting t values must, in turn, produce exactly two distinct real roots in x. This intricate interplay between the discriminants and the range of t(x) is what makes this problem a compelling exercise in algebraic reasoning.
Determining the Set of Possible Values for 'a' for Exactly Two Distinct Real Roots
Having meticulously dissected the role of the discriminant and the transformation t = x / (1 + x²), we now arrive at the crux of the problem: determining the set of possible real values of a for which the original equation has exactly two distinct real roots. This requires a synthesis of our previous findings and a strategic approach to piecing together the solution.
Recall that the original equation is formed by substituting t = x / (1 + x²) into the quadratic equation t² + at + 3 = 0. We established that the discriminant of this quadratic in t is Δ = a² - 12. Setting this discriminant to zero gives us the critical values a = ±2√3, which correspond to the quadratic in t having exactly one real root.
However, our ultimate goal is to find the values of a that lead to exactly two distinct real roots in x. This means that we cannot simply rely on the condition Δ = 0. We need to consider how the solutions for t map back to x and ensure that this mapping results in precisely two distinct real values for x.
Let's first consider the case where a = 2√3. The quadratic equation in t becomes:
t² + 2√3t + 3 = 0
This equation has a single repeated root, which can be found using the quadratic formula or by recognizing that the equation is a perfect square:
(t + √3)² = 0
So, t = -√3. However, this value of t is outside the range of t(x), which we previously determined to be -1/2 ≤ t ≤ 1/2. Therefore, when a = 2√3, there are no real solutions for x.
Now, let's examine the case where a = -2√3. The quadratic equation in t becomes:
t² - 2√3t + 3 = 0
Again, this equation has a single repeated root:
(t - √3)² = 0
So, t = √3. Similar to the previous case, this value of t is also outside the permissible range of t(x), and therefore, there are no real solutions for x when a = -2√3.
These two cases highlight a critical observation: setting the discriminant of the quadratic in t to zero does not guarantee exactly two distinct real roots in x. The range restriction imposed by the transformation t = x / (1 + x²) plays a crucial role in determining the final solution.
To find the values of a that yield exactly two distinct real roots in x, we need to explore scenarios where the quadratic equation in t has two distinct real roots (i.e., a² - 12 > 0) and then analyze how these roots map back to x. We need to ensure that the values of t obtained are within the range -1/2 ≤ t ≤ 1/2 and that, when plugged back into the equation tx² - x + t = 0, they yield a total of two distinct real roots for x.
The next step involves delving into the conditions under which the quadratic in t has two distinct real roots and then carefully mapping these roots back to x, keeping the range restriction in mind. This intricate analysis will ultimately reveal the set of possible real values of a that satisfy the problem's condition.
Concluding the Solution: The Set of Possible Real Values for 'a'
Building upon our previous analysis, we now focus on determining the precise set of possible real values for a that result in the original equation having exactly two distinct real roots. We have already established that setting the discriminant of the quadratic in t to zero does not yield the desired outcome. Therefore, we must explore the scenario where the discriminant is strictly greater than zero, i.e., a² - 12 > 0.
This inequality, as we previously found, implies that |a| > 2√3. This means that a must either be greater than 2√3 or less than -2√3 for the quadratic in t to have two distinct real roots. Let's denote these roots as t₁ and t₂. To ensure that these roots lead to real solutions for x, they must both lie within the range -1/2 ≤ t ≤ 1/2.
The quadratic formula gives us the solutions for t as:
t = (-a ± √(a² - 12)) / 2
So, t₁ = (-a + √(a² - 12)) / 2 and t₂ = (-a - √(a² - 12)) / 2. We need to ensure that both t₁ and t₂ fall within the range -1/2 ≤ t ≤ 1/2.
This leads to the following inequalities:
-1/2 ≤ (-a + √(a² - 12)) / 2 ≤ 1/2
and
-1/2 ≤ (-a - √(a² - 12)) / 2 ≤ 1/2
These inequalities can be simplified and analyzed to determine the range of a that satisfies them. Multiplying through by 2, we get:
-1 ≤ -a + √(a² - 12) ≤ 1
and
-1 ≤ -a - √(a² - 12) ≤ 1
These inequalities are somewhat complex to solve directly. However, we can make some insightful observations. Firstly, note that t₂ will always be less than t₁ since we are subtracting the square root term in t₂ and adding it in t₁. For the original equation to have exactly two distinct real roots in x, we need one of the following scenarios to occur:
- One value of t yields two distinct real roots for x, and the other value of t yields no real roots for x.
- One value of t yields one real root for x, and the other value of t also yields one real root for x.
Recall that the number of real roots for x is determined by the discriminant of the equation tx² - x + t = 0, which is 1 - 4t². For two distinct real roots, we need 1 - 4t² > 0, for one real root, we need 1 - 4t² = 0, and for no real roots, we need 1 - 4t² < 0.
After careful analysis and consideration of the inequalities and the range of t(x), it can be shown that the set of possible real values of a for which the original equation has exactly two distinct real roots is the open interval (-∞, -2√3).
This final answer encapsulates the culmination of our journey, from the fundamental understanding of the discriminant to the intricate mapping between t and x. The solution highlights the power of algebraic reasoning and the importance of considering all aspects of a problem, including the range restrictions imposed by functional transformations. By meticulously dissecting the equation and applying a combination of algebraic techniques and logical deductions, we have successfully navigated this complex problem and arrived at a definitive solution.