Problem 56 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the roots of the equation \(x^{2}-2 a x+a^{2}+a-3=0\) are real and less than 3 , then (A) \(a<2\) (B) \(2 \leq a \leq 3\) (B) \(3 \leq a \leq 4\) (D) \(a>4\)

Step-by-Step Solution

Answer

(A) \(a < 2\)

Step 1: Analyze the Conditions for Real Roots

For the equation \(x^2 - 2ax + a^2 + a - 3 = 0\) to have real roots, the discriminant must be non-negative. Thus, we need \((2a)^2 - 4\cdot 1 \cdot (a^2 + a - 3) \geq 0\). Simplifying, \(4a^2 - 4a^2 - 4a + 12 \geq 0\), which reduces to \( -4a + 12 \geq 0\). Thus, \(a \leq 3\).

Step 2: Ensure Roots Are Less Than 3

We need the roots to satisfy \(x < 3\). First, check the sum of the roots, given by \(2a\), which must satisfy \(\frac{2a}{2} < 3\) or \(a < 3\). Next, check individual root conditions if necessary. However, since both roots are symmetric, it suffices that the sum of the roots be less than 6.

Step 3: Assess All Given Conditions and Options

We found that the roots are real for \(a \leq 3\) and less than 3 for \(a < 3\). Therefore, the only range satisfying both conditions is within option \((A)\) where \(a < 2\). Confirming against the options, \(2 \leq a \leq 3\) works for roots being real but not less than 3, ruling that out.

Key Concepts

Real Roots of Quadratic Equations

For a quadratic equation to have real roots, its discriminant must be a non-negative value. This basically means that the graph of the parabola either touches or intersects the x-axis.
The quadratic equation is generally given as:

• \( ax^2 + bx + c = 0 \)

The discriminant is expressed using the formula: \( b^2 - 4ac \).
If it is zero or greater, the roots are real. Here's what these values imply:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), it has exactly one real root, often called a repeated or double root.
• If \( b^2 - 4ac < 0 \), the roots are complex or imaginary, not real.

Understanding these basics helps solve and analyze quadratic equations effectively.

Understanding the Discriminant

The discriminant \( b^2 - 4ac \) of a quadratic equation acts like a telling indicator for the type of roots. In our given problem, the quadratic equation is \( x^2 - 2ax + a^2 + a - 3 = 0 \).
The coefficients are:

• \( a = 1 \) (leading term)
• \( b = -2a \)
• \( c = a^2 + a - 3 \)

By substituting these into the discriminant formula, we get:

• \( (-2a)^2 - 4 \cdot 1 \cdot (a^2 + a - 3) \)

Which simplifies further to \( -4a + 12 \).
Since the problem asks for real roots, this implies \( -4a + 12 \geq 0 \), resulting in \( a \leq 3 \).
Understanding this helps quickly assess if roots will be real.

Roots Inequality and Their Relationship with Parameters

When analyzing a quadratic equation like \( x^2 - 2ax + a^2 + a - 3 = 0 \), determining not just if roots are real, but also their magnitude relative to a particular value can be challenging.
In our exercise, we needed the roots to be less than 3. This means not only should the discriminant condition \( a \leq 3 \) be satisfied, we also need the averages of the roots to be less than 3.To check inequality constraints, consider the relationships within root and coefficient rules:

• The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \(-b/a \).
• This problem indicates the sum \( 2a \) must satisfy \( 2a/2 < 3 \). Hence, \( a < 3 \).
Combining the real root condition \( a \leq 3 \) and roots less than 3 \( a < 3 \), we get the valid range for \( a < 2 \), ensuring real roots that are all less than 3.