Problem 17 - A Complete Resource Book in Mathematics for JEE Main

Question

Both the roots of the equation \((x-b)(x-c)+(x-a)\) \((x-c)+(x-a)(x-b)=0\) are always (A) positive (B) negative (C) real (D) None of these

Step-by-Step Solution

Answer

The roots are always real (Option C).

Step 1: Expand the Equation

The given equation is \((x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b)=0\). Expand each term to simplify the equation:- \((x-b)(x-c) = x^2 - (b+c)x + bc\)- \((x-a)(x-c) = x^2 - (a+c)x + ac\)- \((x-a)(x-b) = x^2 - (a+b)x + ab\).

Step 2: Combine Like Terms

Combine all expanded terms:1. \(x^2 - (b+c)x + bc + x^2 - (a+c)x + ac + x^2 - (a+b)x + ab = 0\)2. Combining like terms gives: \(3x^2 - (2a + 2b + 2c)x + (ab + ac + bc) = 0\).

Step 3: Analyze the Resulting Quadratic

The equation simplifies to \(3x^2 - 2(a + b + c)x + (ab+ac+bc) = 0\). It's a quadratic in the form \(ax^2 + bx + c = 0\) with \(a = 3\), \(b = -2(a+b+c)\), and \(c = ab+ac+bc\).

Step 4: Determine the Nature of the Roots

The discriminant for a quadratic \(ax^2 + bx + c = 0\) is \(b^2 - 4ac\). Calculate the discriminant:- \((-2(a+b+c))^2 - 4 \times 3 \times (ab+ac+bc) = 4(a+b+c)^2 - 12(ab+ac+bc)\).- This simplifies to \(4[(a^2 + b^2 + c^2) - (ab + ac + bc)] = 0\), implying the discriminant can be varied but assumes a positive or zero value, ensuring real roots.

Step 5: Conclusion on the Roots

The discriminant we analyzed is either zero or positive, meaning the roots are always real. Therefore, option (C) is the correct answer.

Key Concepts

Discriminant and Nature of the Roots

The discriminant is a crucial part of any quadratic equation. It helps us understand the nature of the roots without actually solving the equation. For a quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated as \(b^2 - 4ac\). This expression plays a significant role in determining the type of roots the equation will have.

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, also known as a repeated or double root.
• If \(D < 0\), the roots are complex and not real.

In our particular exercise, we determined that the discriminant simplifies to a non-negative value \(4[(a^2 + b^2 + c^2) - (ab + ac + bc)]\). This result implies that the roots are either distinct real numbers or a repeated real root, ensuring the roots are always real.

Real Roots

The concept of real roots boils down to whether the solutions of a quadratic equation can be plotted on a standard number line. Real roots are simply the values of \(x\) which will satisfy the equation, making the left-hand side equal to zero. Remember, for an equation to have real roots, its solutions must be values that we can "see" or "understand" within the realm of real numbers.

In our exercise, we've explored the quadratic expression \(3x^2 - 2(a+b+c)x + (ab+ac+bc) = 0\). Given the previously calculated non-negative discriminant, we confidently conclude that the equation has real roots. This is a fundamental confirmation that no matter the values of \(a\), \(b\), and \(c\), they will always lead to actual solutions on a number line.

Simplification of Quadratic Equations

Simplifying a quadratic equation often begins with expanding and rearranging terms to make the equation easier to handle. In the provided exercise, the expression `(x-b)(x-c)+(x-a)(x-c)+(x-a)(x-b) = 0` was systematically expanded and combined into the more recognizable quadratic form \(3x^2 - 2(a + b + c)x + (ab + ac + bc) = 0\).

• Expansion involves multiplying terms such as \((x-b)(x-c)\) to \(x^2 - (b+c)x + bc\).
• Combining like terms simplifies the expression further, turning multiple smaller parts into a single, neat equation.
• The simplified form makes it easier to examine the equation for relevant properties, such as the discriminant.

By organizing the equation into \(ax^2 + bx + c = 0\), it is less complex and ready for further analysis, allowing us to determine the nature of its roots accurately.