Problem 87 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+c x+a b=0\) have a common root, then show that their other roots are the roots of the equation \(x^{2}+a x+b c=0\).

Step-by-Step Solution

Answer

The short answer is that the other roots of the given equations are the roots of the equation \(x^2 + ax + bc = 0\). This result is obtained by using Vieta's Formulas to find a relationship between the coefficients of the two given equations and then using this relationship to find the equation with roots 𝛽 and 𝛾.

Step 1: Identify the common root

Let's denote the common root of the two given equations as α. The first equation is \(x^2 + bx + ca = 0\) with roots α and β. The second equation is \(x^2 + cx + ab = 0\) with roots α and γ.

Step 2: Use Vieta's Formulas

Since α and β are roots of the first equation, we can use Vieta's Formulas to express the sum and product of the roots. Sum of the roots: \(\alpha + \beta = -b\) Product of the roots: \(\alpha \beta = ca\) Similarly, for the second equation with roots α and γ: Sum of the roots: \(\alpha + \gamma = -c\) Product of the roots: \(\alpha \gamma = ab\)

Step 3: Find a relationship between the coefficients

Now we can find a relationship between the coefficients of the two given equations. Subtract the sum of the roots formula from both equations: \(\beta - \gamma = \alpha - b - (-c) = \alpha + c - b\) Multiply the product of the roots formula from both equations: \((\alpha \beta)(\alpha \gamma) = (ca)(ab)\) \(\alpha^2 \beta \gamma = cab^2\)

Step 4: Use the relationship to find the equation with roots β and γ

To find the quadratic equation with roots β and γ, we will use the relationship between the coefficients and the sum and product of the roots β and γ. Sum of the roots: \(\beta + \gamma = -a\) Product of the roots: \(\beta \gamma = \frac{cab^2}{\alpha^2}\) Using Vieta's Formulas backward, we find the equation (denoted as \(f(x)=0\)) as: \(f(x) = x^2 - ax + \frac{cab^2}{\alpha^2}\) Now, substituting α + c - b for β - γ in the sum of roots formula: \(-a = (\alpha + c - b) + (-\alpha) = -b + c\) From here, we find that \(a = b - c\). Substituting this in the equation obtained above, we have: \(f(x) = x^2 - (b - c)x + \frac{cab^2}{\alpha^2}\) The product of the roots of this equation is \(\frac{cab^2}{\alpha^2}\). Since α is the common root of the two given equations, its square will be their common part. So, the ratio of the product of the roots of the equation \(f(x)=0\) will be: \(bc = \frac{cab^2}{\alpha^2}\)

Step 5: Final result

Therefore, the other roots of the given equations are the roots of the equation: \(x^2 - (b - c)x + bc = 0\) or \(x^2 + ax + bc = 0\) Hence, the other roots of the given equations are the roots of the equation \(x^2 + ax + bc = 0\), as required.

Key Concepts

Vieta's Formulas

Vieta's Formulas are a set of equations that relate the coefficients of a polynomial to the sums and products of its roots. These formulas can greatly simplify solving polynomial equations where the relationship between roots is needed. Vieta's Formulas are mainly used for quadratic equations of the form \(x^2 + bx + c = 0\), relating the roots \(\alpha\) and \(\beta\) as follows:

• The sum of the roots \(\alpha + \beta = -b\).
• The product of the roots \(\alpha \beta = c\).

In the context of the given problem, we use Vieta's Formulas to express the roots of two quadratic equations and the coefficients. When a root \(\alpha\) is common to both equations, Vieta's Formulas allow us to derive essential relationships that help identify additional roots or confirm existing ones. Essentially, these formulas help us link the algebraic expressions to the visual representation of roots on a graph.

Common Roots

Having a common root means that two different equations share at least one root. In the problem above, the two quadratic equations share a common root, \(\alpha\). For these specific equations, this implies a deeper connection between them, governed by the values of their coefficients.To identify and work with common roots, one would:

• Determine which root is shared, like \(\alpha\) in this case.
• Use this shared root in Vieta’s Formulas to establish relationships or constraints between equated sums and products, which makes it easier to understand the structure of the roots.

Understanding common roots is vital because they often signify symmetries or elegant relationships between different equations. Such insights can simplify the complexity around solving these simultaneous expressions.

Coefficient Relationships

The coefficients in a polynomial give a wealth of information about the nature of its roots. In the exercise provided, we see that the coefficients \(b, c, a\) of the quadratic equations are connected through a shared root situation.Given the two quadratic equations \(x^2+bx+ca=0\) and \(x^2+cx+ab=0\), the presence of a common root leads to equations involving these coefficients. For example:

• Through substituting and rearranging, insights into how \(b, c, a\) relate to the roots, go beyond just Vieta's sums and products.
• This relationship enables us to derive a third equation whose roots are directly influenced by these coefficients and the commonality of \(\alpha\).

By understanding these relationships, you modify one equation to help find solutions to another. This approach can also indicate properties such as whether the polynomial is factorable, or what the potential nature and behavior of the roots will be. Thus, coefficients are more than mere numbers—they are key indicators of the mathematical symmetries involved in polynomial equations.