Problem 136 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: If the roots of the equations \(x^{2}-b x+c=0\) and \(x^{2}-c x+b=0\) differ by the same quantity, then \(b+c\) is equal to \(-4\). Reason: If \(\alpha, \beta\) are the roots of the equation \(A x^{2}+\) \(B x+C=0\), then \(\alpha-\beta=\frac{\sqrt{B^{2}-4 A C}}{A}\)

Step-by-Step Solution

Answer

The correct answer is that \(b + c = -4\).

Step 1: Identify the roots

For the equation \(x^2 - bx + c = 0\), use the quadratic formula to find the roots: \(x = \frac{b \pm \sqrt{b^2 - 4c}}{2}\).Similarly, for the equation \(x^2 - cx + b = 0\), the roots are: \(x = \frac{c \pm \sqrt{c^2 - 4b}}{2}\).

Step 2: Difference of roots for each equation

For the first equation, the difference in roots \(\alpha - \beta\) is:\[\frac{\sqrt{b^2 - 4c}}{1} = \sqrt{b^2 - 4c}\]For the second equation, the difference in roots \(\gamma - \delta\) is:\[\frac{\sqrt{c^2 - 4b}}{1} = \sqrt{c^2 - 4b}\]

Step 3: Equate the differences of roots

Given that the roots differ by the same quantity, equate the two expressions:\[\sqrt{b^2 - 4c} = \sqrt{c^2 - 4b}\]

Step 4: Square both sides

By squaring both sides, we have:\[b^2 - 4c = c^2 - 4b\]

Step 5: Rearrange the equation

Rearrange the equation to bring all terms to one side:\[b^2 - c^2 = 4c - 4b\]Which simplifies to:\[(b - c)(b + c) = 4(c - b)\]

Step 6: Solve for \(b + c\)

Factor out \((b - c)\) from both sides:\[(b - c)(b + c + 4) = 0\]This gives two possible solutions: \(b - c = 0\) or \(b + c + 4 = 0\).Since we are asked about \(b + c\), solve \(b + c + 4 = 0\) to find:\[b + c = -4\].

Key Concepts

Roots of Quadratic Equation

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). They are called 'quadratic' because 'quad' means square, referring to the highest power of the variable, which is two. Understanding the roots of these equations is crucial in solving them.
To find the roots of a quadratic equation, we can use the quadratic formula. The roots \(\alpha\) and \(\beta\) are given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The term under the square root, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots:

• If it's positive, there are two distinct real roots.
• If zero, there's exactly one real root, also referred to as a repeated or double root.
• If negative, the roots are complex or imaginary.

For our equation \( x^2 - bx + c = 0 \), we can substitute \( a = 1 \), \( b = -b \), and \( c = c \) into the formula to find its roots. This formula helps in solving any quadratic equation efficiently.

Difference of Roots

The difference between the roots of a quadratic equation is an important concept. It helps us gain insights into the behavior or symmetry of the polynomial's graph. For the equation \( Ax^2 + Bx + C = 0 \), this difference can be expressed mathematically. Given roots \( \alpha \) and \( \beta \), the difference \( \alpha - \beta \) is calculated as:

• \( \frac{\sqrt{B^2 - 4AC}}{A} \)

This expression comes from the quadratic formula and tells us how far apart the roots of the equation are on a graph.
In our original problem, we dealt with two quadratic equations whose roots differed by the same amount. This common difference was used to derive an equation. By setting their differences equal, we could solve for specific parameters like \( b + c \). The solution steps involved setting \( \sqrt{b^2 - 4c} = \sqrt{c^2 - 4b} \), squaring both sides, and manipulating the expressions to reach a valid solution for \( b + c \). This illustrates how differences in roots guide us toward finding or verifying specific values in quadratic equations.

Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations. It provides a straightforward method for finding the roots of any quadratic equation, whether they are real or complex. This formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Each part of this formula plays a critical role in determining the roots:

• \( -b \) represents the sum of the roots times -1 due to the equation's symmetry around the y-axis.
• The \( \pm \) symbol indicates two possible roots, representing the equation's parabolic nature.
• The square root part, \( \sqrt{b^2 - 4ac} \), as mentioned earlier, is the discriminant, which tells us about the nature of the roots.

For equations like \( x^2-bx+c = 0 \) and \( x^2-cx+b = 0 \), as in our problem, the quadratic formula was directly applied by substituting the coefficients. It allowed us to calculate and compare the differences in the roots accurately. The formula is a reliable tool in algebra and beyond for simplifying and resolving calculations related to quadratic equations efficiently.