Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\alpha\) and \(\beta(\alpha<\beta)\), are the roots of the equation \(x^{2}+\) \(b x+c=0\), where \(c<0

Step-by-Step Solution

Answer

The correct option is (D): \(\alpha < 0 < |\alpha| < \beta\).

Step 1: Recognize properties of roots

The roots of a quadratic equation \(ax^2 + bx + c = 0\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\). According to Viète's formulas, we have two primary conditions: the sum of the roots \(\alpha + \beta = -b\), and the product of the roots \(\alpha\beta = c\). With \(b > 0\) and \(c < 0\), we need to examine these conditions further.

Step 2: Consider the sum of roots

The sum of the roots \(\alpha + \beta = -b\) is negative since \(b > 0\). This means that the sum \(\alpha + \beta\) is negative, suggesting at least one of the roots is negative.

Step 3: Consider the product of roots

The product of the roots \(\alpha\beta = c\) is negative because \(c < 0\). This further implies that one root is positive and one is negative as a product of two numbers is negative only when one is positive and the other is negative.

Step 4: Determine which root is negative and which is positive

Since \(\alpha + \beta\) is negative (from Step 2) and \(\alpha\beta\) is negative (from Step 3), let \(\alpha < 0\) and \(\beta > 0\). This implies that the negative root is \(\alpha\) and the positive root is \(\beta\).

Step 5: Check options

Finally, we validate based on the conditions we determined: \(\alpha < 0 < \beta\). Among the options, (D) \(\alpha < 0 < |\alpha| < \beta\) aligns well because for the condition \(|\alpha| < \beta\) to hold true, \(\alpha\) should be negative and its absolute value smaller than \(\beta\), which is consistent with \(\alpha < 0\).

Key Concepts

Roots of quadratic equations

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the solutions are commonly known as the roots. They are typically represented by \(\alpha\) and \(\beta\), where for this particular problem, it's given that \(\alpha < \beta\). The quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{2a}\) helps you find these roots. After calculating, you identify \(\alpha\) and \(\beta\) by comparing their values.

In this case, the exercise provides a scenario where \(b > 0\) and \(c < 0\). This setup is crucial because:

• If the sum of the roots \(\alpha + \beta = -b\) is negative, at least one of the roots must also be negative, given that \(b\) is positive.
• The product of the roots \(\alpha\beta = c\) being negative implies that the roots must be of opposite signs.

These observations about the roots help us determine their nature: one positive and the other negative, leading us to further conclusions in the exercise.

Viète's formulas

Viète's formulas are a set of relationships between the coefficients of a polynomial and its roots. Specifically, for a quadratic equation \(ax^2 + bx + c = 0\), the formulas are:

• The sum of the roots: \(\alpha + \beta = -\frac{b}{a}\).
• The product of the roots: \(\alpha\beta = \frac{c}{a}\).

For the given exercise, these formulas are very powerful as they help us understand the signs of the roots without necessarily solving the entire quadratic equation. Given \(b > 0\) and \(c < 0\):

- The sum of the roots \(-\frac{b}{1} = -b\) tells us it is negative, as expected from a positive \(b\). This implies the roots' sum is negative, requiring at least one root to be negative for compensation.
- The product \(\frac{c}{1} = c\) being negative directly informs us that one root is negative and the other is positive. Thus, Viète's formulas provide a shortcut to predict the nature of the roots.

Negative and positive roots

When examining a quadratic equation with one positive and one negative root, their behavior offers deeper insights into the equation. Let's illustrate the scenario as per the exercise:
- \(\alpha\) is negative and \(\beta\) is positive.
- At first glance, it could seem tricky figuring out whether \(|\alpha|\) is smaller or larger than \(\beta\). However, analyzing further indicates:

• The positive root \(\beta\) must be greater in magnitude (or absolute value) than the negative \(|\alpha|\), meaning \(|\alpha| < \beta\).

This finding flips the intuitive expectation because the position on the number line gives \(\beta\) a larger numerical weight than what \(\alpha\) offers in negativity if their sum is to remain negative overall.

This clarity resolves the answer to the exercise as option (D): \(\alpha < 0 < |\alpha| < \beta\). Here, the negative and positive root perspectives demonstrate a critical part of understanding quadratic roots' nature.