Parametric Quadratics & Discriminant Analysis

The Graphical Toolkit for Locating Roots

Consider the quadratic equation $f(x)=a{x}^2+bx+c=0$. The core of this topic is understanding the graph of the corresponding parabola. For consistency, we will always work with the assumption that the leading coefficient $a>0$, meaning the parabola opens upwards. If you encounter a problem with $a<0$, simply multiply the entire equation by -1 to make the leading coefficient positive (this does not change the roots).

The key features of the parabola that we will use are:

1. The Discriminant ($D$): Determines if the parabola intersects the x-axis ($D\ge 0$) or not ($D<0$).
2. The Vertex's x-coordinate ($-b/2a$): The axis of symmetry of the parabola.
3. The Value of the function at a point $k$ ($f(k)$): Tells us if the point $(k,f(k))$ is above ($f(k)>0$), below ($f(k)<0$), or on ($f(k)=0$) the x-axis.

Let's break down the most common scenarios into a systematic set of conditions.

Scenario 1: Both Roots are Greater than a Number $d$

For both roots, $\alpha$ and $\beta$, of $a{x}^2+bx+c=0$ to be greater than a number $d$ (with $a>0$), three conditions must be met simultaneously:

1. $D\ge 0$: The roots must be real. (We use $\ge$ to include the case of equal roots).
2. $-b/2a>d$: The vertex (the midpoint of the roots) must be to the right of $d$.
3. $f(d)>0$: The value of the function at $d$ must be positive, ensuring $d$ is not between the roots.

The Unacademy notes on Quadratic Equations provide an excellent walkthrough of this case with a solved example.

Scenario 2: A Number $d$ Lies Between the Roots

This is a simpler but very important case. For a number $d$ to lie between the roots $\alpha$ and $\beta$ (i.e., $\alpha <d<\beta$), only one condition is needed (assuming $a>0$):

1. $f(d)<0$

If the parabola opens upwards and its value is negative at $x=d$, it must cross the x-axis on both sides of $d$ to go back up. This single condition cleverly implies that $D>0$ (two distinct real roots exist) and that they are on either side of $d$.

Scenario 3: Both Roots Lie in an Interval $(d,e)$

This is a combination of the logic from Scenario 1. For both roots to be between $d$ and $e$ (with $a>0$), we need to ensure the roots are real, the vertex is within the interval, and the function is positive at both $d$ and $e$.

1. $D\ge 0$
2. $f(d)>0$
3. $f(e)>0$
4. $d<-b/2a<e$

The following resource provides a clear example of this scenario.

Scenario 4: Exactly One Root Lies in an Interval $(d,e)$

This is a classic JEE problem type with a subtle trap. The most straightforward condition is that the function values at the endpoints have opposite signs:

• $f(d)\cdot f(e)<0$

This means one endpoint is above the x-axis and one is below, guaranteeing the parabola crosses the axis exactly once between them.

The Trap: What if one of the roots is exactly $d$ or $e$? The condition $f(d)\cdot f(e)<0$ does not account for this. A complete solution must also check the cases where $f(d)=0$ or $f(e)=0$. If, for example, $f(d)=0$, you must then check if the other root lies within the interval $(d,e)$.

The Unacademy notes handle this perfectly.

Practice Problems

Now, let's apply this framework to some JEE-level problems.

Problem 1: Find the number of integral values of m for which the quadratic expression (1 + 2m)x^2 – 2(1 + 3m)x + 4(1 + m) is always positive for all $x\in \mathbb{R}$.

• Hint: For a quadratic $a{x}^2+bx+c$ to be always positive, the parabola must open upwards and never touch or cross the x-axis. What does this imply for the leading coefficient a and the discriminant D? You can find a similar problem worked out in the Allen problem set.

Problem 2: Find the set of all integral values of c for which one root of the equation (c–5)x^2–2cx + (c–4) = 0 lies in the interval (0,2) and its other root lies in the interval (2,3).

• Hint: This means the number 2 lies between the roots. Additionally, the roots are separated by the numbers 0 and 3. What does this imply about the signs of $f(0)$, $f(2)$, and $f(3)$?
• Specifically, 0 and 2 must contain one root, and 2 and 3 must contain the other. This translates to two conditions: $f(0)f(2)<0$ and $f(2)f(3)<0$. Notice that the leading coefficient (c-5) is a parameter itself. You may need to consider cases for its sign, or you can find a way to combine the inequalities that makes the leading coefficient's sign irrelevant.

Conclusion

Mastering the location of roots is the final piece of the puzzle for the chapter on quadratic equations. It is a topic that beautifully blends algebraic manipulation with graphical intuition.

Key Takeaways:

• Problems involving the location of roots of a parametric quadratic equation are solved by setting up a system of inequalities for the parameter.
• Always begin by ensuring the leading coefficient is positive ($a>0$).
• The essential tools are the discriminant ($D$), the vertex's x-coordinate ($-b/2a$), and the function's value at the interval boundaries ($f(k)$).
• Be mindful of subtleties, especially the difference between strict inequalities (intervals like (d,e)) and non-strict ones (intervals like [d,e]), and the special cases where a root might coincide with an interval boundary.

This concludes our intensive study of Quadratic Equations. In the next lesson, we will begin the second module of our course: Sequences and Series. We'll start with Arithmetic Progressions (AP), revisiting concepts you've seen before but with the depth and problem-solving focus required for the JEE Advanced.