Problem 99 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the set of values of \(p\) for which the roots of the equation \(3 x^{2}+2 x+p(p-1)=0\) are of opposite signs.

Step-by-Step Solution

Answer

The roots of the equation \(3x^2 + 2x + p(p - 1) = 0\) have opposite signs if \(0

Step 1: Identify the quadratic equation

Given quadratic equation is \(3x^2 + 2x + p(p - 1) = 0\).

Step 2: Find Expressions for the roots

Let the roots of this quadratic equation be \(\alpha\) and \(\beta\). We can use the Vieta's relations to represent the roots in terms of the coefficients. Sum of roots: \(\alpha + \beta = -\frac{2}{3}\) Product of roots: \(\alpha\beta = \frac{p(p - 1)}{3}\)

Step 3: Determine the conditions for opposite signs of the roots

Since one root is positive and the other root is negative, their sum \(\alpha + \beta\) will be negative. But we already know their sum is \(-\frac{2}{3}\), which is negative, so the given condition on the roots already holds true. Now, let's focus on the product of the roots: Since one root is positive and the other is negative, their product \(\alpha\beta\) will be negative. We have, \(\alpha\beta = \frac{p(p - 1)}{3}\).

Step 4: Find the values of p that result in a negative product of roots

Since we want the product of roots to be negative, we must have: \(\frac{p(p-1)}{3} < 0\) We can analyze the inequality by considering the signs of the factors of the inequality: 1. If \(p-1>0 \Rightarrow p>1\), then both the factors \(p\) and \(p - 1\) are positive, so the product is positive, and the inequality doesn't hold. 2. If \(0

Step 5: Conclusion

Based on our analysis, the roots \(\alpha\) and \(\beta\) are of opposite signs if \(0

Key Concepts

Vieta's relations

Vieta's relations are incredibly useful tools when dealing with quadratic equations. These formulas relate the coefficients of a polynomial to sums and products of its roots. Specifically for a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's relations tell us that the sum of roots \(-\frac{b}{a}\), and the product of the roots \(\frac{c}{a}\). Taking the example from the exercise, where our quadratic equation is \(3x^2 + 2x + p(p - 1) = 0\), the roots can be expressed using Vieta's relations as:

• Sum of roots (\(\alpha + \beta\)): \(-\frac{b}{a} = -\frac{2}{3}\)
• Product of roots (\(\alpha\beta\)): \(\frac{c}{a} = \frac{p(p - 1)}{3}\)

When analyzing problems that require understanding the nature of the roots, like whether they have opposite signs, Vieta's formulas become particularly handy. They allow us to set up equations and inequalities that can lead us to the set of values (in this case, values for \(p\)) that satisfy the conditions imposed on the roots.

Inequalities

Inequalities are statements about the relative size or order of two objects. They allow us to compare and establish the relationships between various values or expressions. Common inequality signs include \(<\), \(>\), \(\leq\), and \(\geq\), representing 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. In the exercise we're analyzing, we determined that for the roots of the given quadratic equation to have opposite signs, their product must be negative. That led to the inequality \(\frac{p(p-1)}{3} < 0\).

To solve such an inequality, we look for the range of values that make the inequality true. We can split it into intervals based on the sign changes in the factors involved. So when you encounter an inequality in the context of roots of a quadratic equation, pay close attention to the factors of the product and where they change signs to determine the solution set.

Quadratic Equation

The quadratic equation, recognizable by its standard form \(ax^2 + bx + c = 0\), is a fundamental element of algebra that surfaces in a variety of mathematical contexts.

Finding the Roots

There are several methods to find the roots of a quadratic equation including factoring, completing the square, and using the quadratic formula. Using these methods or applying Vieta's relations, we can determine the roots' exact values or their relationship to the coefficients of the equation.

Conditions on Roots

Sometimes, we're interested not just in the actual roots, but in their characteristics, like in the exercise which required the roots to have opposite signs. In such cases, understanding the nature of the quadratic expression and using inequalities together are key to finding the solution. For instance, if we know that a quadratic equation's roots have opposite signs, we can reliably deduce that the quadratic equation's constant term (in the standard form) and its product of roots must be negative. In summary, mastering the quadratic equation involves not just solving for the roots, but also interpreting their properties and how they relate to the equation's coefficients.

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Problem 100

Other exercises in this chapter

Problem 97

For what values of \(a\) does the function \(x^{2}+2(a+1) x+9 a-5\) has i. no real roots. ii. only negative roots. iii. only positive roots.

View solution

Problem 98

If \(b, c>0\), then show that roots of the equation \(x^{2}+b x-c=0\) are of opposite sign.

View solution

Problem 100

If the roots of the equation \(3 x^{2}+2\left(k^{2}+1\right) x+\left(k^{2}-3 k+2\right)=0\) be of opposite signs, then prove that \(1

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Problem 101

For what values of \(a\) does the function \(2 x^{2}-\left(a^{3}+8 a-1\right) x+a^{2}-4 a\) has the roots of opposite signs?

View solution

Key Concepts

Vieta's relations

Vieta's relations are incredibly useful tools when dealing with quadratic equations. These formulas relate the coefficients of a polynomial to sums and products of its roots. Specifically for a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's relations tell us that the sum of roots \(-\frac{b}{a}\), and the product of the roots \(\frac{c}{a}\). Taking the example from the exercise, where our quadratic equation is \(3x^2 + 2x + p(p - 1) = 0\), the roots can be expressed using Vieta's relations as:

• Sum of roots (\(\alpha + \beta\)): \(-\frac{b}{a} = -\frac{2}{3}\)
• Product of roots (\(\alpha\beta\)): \(\frac{c}{a} = \frac{p(p - 1)}{3}\)

When analyzing problems that require understanding the nature of the roots, like whether they have opposite signs, Vieta's formulas become particularly handy. They allow us to set up equations and inequalities that can lead us to the set of values (in this case, values for \(p\)) that satisfy the conditions imposed on the roots.

Inequalities

Inequalities are statements about the relative size or order of two objects. They allow us to compare and establish the relationships between various values or expressions. Common inequality signs include \(<\), \(>\), \(\leq\), and \(\geq\), representing 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. In the exercise we're analyzing, we determined that for the roots of the given quadratic equation to have opposite signs, their product must be negative. That led to the inequality \(\frac{p(p-1)}{3} < 0\).

To solve such an inequality, we look for the range of values that make the inequality true. We can split it into intervals based on the sign changes in the factors involved. So when you encounter an inequality in the context of roots of a quadratic equation, pay close attention to the factors of the product and where they change signs to determine the solution set.

Quadratic Equation

The quadratic equation, recognizable by its standard form \(ax^2 + bx + c = 0\), is a fundamental element of algebra that surfaces in a variety of mathematical contexts.

Finding the Roots

There are several methods to find the roots of a quadratic equation including factoring, completing the square, and using the quadratic formula. Using these methods or applying Vieta's relations, we can determine the roots' exact values or their relationship to the coefficients of the equation.

Conditions on Roots

Sometimes, we're interested not just in the actual roots, but in their characteristics, like in the exercise which required the roots to have opposite signs. In such cases, understanding the nature of the quadratic expression and using inequalities together are key to finding the solution. For instance, if we know that a quadratic equation's roots have opposite signs, we can reliably deduce that the quadratic equation's constant term (in the standard form) and its product of roots must be negative. In summary, mastering the quadratic equation involves not just solving for the roots, but also interpreting their properties and how they relate to the equation's coefficients.