If α and β are the roots of the quadratic equation 2x 2+ 6x + k = 0, where k < 0, then what is the maximum value of (α/β + β/α)?
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Analyzing the Quadratic Equation and its Roots
We are given the quadratic equation \(2x^2 + 6x + k = 0\), where \(k < 0\). Let the roots of this equation be \(\alpha\) and \(\beta\). According to Vieta's formulas, we can relate the coefficients of the quadratic equation to the sum and product of its roots:
- Sum of roots: \(\alpha + \beta = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{6}{2} = -3\)
- Product of roots: \(\alpha \beta = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{k}{2}\)
We are given that \(k < 0\). This means the product of the roots, \(\alpha \beta = k/2\), is also negative. If the product of two numbers is negative, the numbers must have opposite signs. Therefore, one root is positive and the other is negative.
Simplifying the Expression to be Maximized
The expression we need to find the maximum value of is \(E = \frac{\alpha}{\beta} + \frac{\beta}{\alpha}\). To simplify this expression, we can find a common denominator: \[E = \frac{\alpha^2 + \beta^2}{\alpha \beta}\] We know that \(\alpha^2 + \beta^2\) can be expressed in terms of the sum and product of the roots using the identity \((\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2\), which gives \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\).
Substitute the values from Vieta's formulas: \(\alpha + \beta = -3\) \(\alpha \beta = k/2\) \[\alpha^2 + \beta^2 = (-3)^2 - 2\left(\frac{k}{2}\right) = 9 - k\] Now substitute this back into the expression for \(E\): \[E = \frac{9 - k}{k/2} = \frac{2(9 - k)}{k} = \frac{18 - 2k}{k}\] This expression can be further simplified by dividing both terms in the numerator by \(k\): \[E = \frac{18}{k} - \frac{2k}{k} = \frac{18}{k} - 2\] So, the expression we need to maximize is \(E = \frac{18}{k} - 2\).
Analyzing the Function of k
We need to find the maximum value of \(E(k) = \frac{18}{k} - 2\) under the condition that \(k < 0\). This means \(k\) can take any value in the interval \((-\infty, 0)\). Let's analyze the behavior of \(E(k)\) as \(k\) varies within this interval.
- As \(k\) approaches \(0\) from the negative side (\(k \to 0^-\)), the term \(\frac{18}{k}\) becomes a large negative number (approaching \(-\infty\)). Therefore, \(E(k) = \frac{18}{k} - 2\) approaches \(-\infty - 2 = -\infty\).
- As \(k\) approaches \(-\infty\) (becomes a very large negative number), the term \(\frac{18}{k}\) approaches \(0\) from the negative side. Therefore, \(E(k) = \frac{18}{k} - 2\) approaches \(0 - 2 = -2\).
The function \(E(k) = \frac{18}{k} - 2\) is strictly increasing as \(k\) increases towards \(0\) within the interval \((-\infty, 0)\). To see this, consider the derivative with respect to \(k\): \[\frac{dE}{dk} = \frac{d}{dk}\left(\frac{18}{k} - 2\right) = 18 \cdot (-k^{-2}) = -\frac{18}{k^2}\] For any \(k < 0\), \(k^2 > 0\), so \(-\frac{18}{k^2} < 0\). This means the function \(E(k)\) is strictly decreasing as \(k\) decreases (moves towards \(-\infty\)) and strictly increasing as \(k\) increases (moves towards \(0\)).
Determining the Range and Maximum Value
Based on the analysis of \(E(k)\) for \(k \in (-\infty, 0)\):
- As \(k \to -\infty\), \(E(k) \to -2\).
- As \(k \to 0^-\), \(E(k) \to -\infty\).
The range of possible values for \(E = \frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) is the interval \((-\infty, -2)\). The question asks for the maximum value of the expression. A set with range \((-\infty, -2)\) does not have a maximum element because it is unbounded above and the upper bound \(-2\) is not included in the set. However, \(-2\) is the least upper bound (supremum) of the set of possible values. This value is approached as \(k\) becomes a large negative number. Given the options provided, \(-2\) is the only value that lies on the boundary of the range \((-\infty, -2)\) and is the supremum. In contexts like this, the supremum is often considered the "maximum value" among the given options if a true maximum is not attained within the feasible region. Therefore, based on the options, the maximum value is \(-2\). This value is approached as \(k \to -\infty\).
| Value of \(k\) | Value of \(E = \frac{18}{k} - 2\) | Behavior |
|---|---|---|
| \(k \to -\infty\) | \(E \to -2\) | Approaching the supremum |
| \(k = -100\) | \(E = -0.18 - 2 = -2.18\) | Value in the range |
| \(k = -10\) | \(E = -1.8 - 2 = -3.8\) | Value in the range |
| \(k = -1\) | \(E = -18 - 2 = -20\) | Value in the range |
| \(k \to 0^-\) | \(E \to -\infty\) | Approaching negative infinity |
The table shows how the value of the expression changes with \(k\). As \(k\) decreases (becomes more negative), \(E\) increases and approaches \(-2\). The maximum value approached is \(-2\).
Revision Table: Quadratic Equation Roots Analysis
| Concept | Description |
|---|---|
| Quadratic Equation | An equation of the form \(ax^2 + bx + c = 0\) where \(a \neq 0\). |
| Roots of Quadratic Equation | The values of \(x\) that satisfy the equation, also known as solutions or zeros. Can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). |
| Vieta's Formulas | Relate the coefficients of a polynomial to the sums and products of its roots. For \(ax^2 + bx + c = 0\), sum of roots is \(-b/a\), product of roots is \(c/a\). |
| Discriminant (\(\Delta\)) | The value \(b^2 - 4ac\) for a quadratic equation. It determines the nature of the roots: \(\Delta > 0\): Two distinct real roots. \(\Delta = 0\): One real root (repeated root). \(\Delta < 0\): Two complex conjugate roots. |
Additional Information: Maximum and Supremum Concepts
In mathematics, the maximum value of a set is the largest element in the set. The supremum (or least upper bound) of a set is the smallest number that is greater than or equal to all elements in the set.
If a set has a maximum element, the maximum is also the supremum. However, a set can have a supremum without having a maximum element. For example, the set \((-\infty, -2)\) does not have a maximum value because no single number is the largest. For any number in the set, there is always a larger number also in the set (closer to \(-2\)). The supremum of the set \((-\infty, -2)\) is \(-2\), which is not an element of the set.
In this problem, the set of possible values for the expression is \((-\infty, -2)\). The maximum value is not attained within the allowed range of \(k\). However, the supremum of this set is \(-2\). Given that \(-2\) is one of the options, it is the most appropriate answer, representing the value that the expression gets arbitrarily close to, and is the least upper bound for the set of values.
Important Questions from Quadratic Equations
If k = c, then the roots of the equation are:
If \(\rm {k}=\frac{{c}}{2},({c} \neq 0)\), then the roots of the equation are :
What is the number of real roots of the equation?
What is the sum of all the roots of the equation?
If α and β are the distinct roots of equation x2 - x + 1 = 0, then what is the value of \(\left|\frac{\alpha^{100}+\beta^{100}}{\alpha^{100}-\beta^{100}}\right|\) ?
