Problem 64 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If one root of the equation \(x^{2}+p x+12=0\) is 4, while the equation \(x^{2}+p x+q=0\) has equal roots, then find the value of \(q\).

Step-by-Step Solution

Answer

The value of q is \(\frac{49}{4}\).

Step 1: Find the value of p using the root of the first equation

Since one root of the equation \(x^2 + px + 12 = 0\) is 4, we can substitute it into the equation: \[4^2 + 4p + 12 = 0\] \[16 + 4p + 12 = 0\] Now, we can solve for p: \[4p = -(16 + 12)\] \[4p = -28\] \[p = -7\]

Step 2: Determine the discriminant of the second equation

Since the second equation \(x^2 + px + q = 0\) has equal roots, its discriminant must be equal to zero. The discriminant of a quadratic equation in the form of \(ax^2+bx+c=0\) is given by: \[D = b^2 - 4ac\] In the case of the second equation, a = 1, b = p and c = q. So, we have: \[D = p^2 - 4q\] As we have already found the value of p: \[D=(-7)^2-4q\] Because the second equation has equal roots, the discriminant must be zero: \[(-7)^2 - 4q = 0\]

Step 3: Solve for the value of q

Now, we can solve for q: \[49 - 4q=0\] \[4q = 49\] \[q = \frac{49}{4}\] Hence, the value of q is \(\frac{49}{4}\).

Key Concepts

Equal Roots

When we talk about equal roots in quadratic equations, it means that a quadratic equation has a specific trait where both solutions or roots are exactly the same. This characteristic happens only under certain conditions. If you stumble upon a quadratic equation in the form of \( ax^2 + bx + c = 0 \) and it has equal roots, it implies that there's just one unique real number that satisfies the equation twice.

A quadratic equation will have equal roots if its discriminant is exactly zero. This singular result affects the nature of its solution, making sure that the curves of a quadratic graph barely touch the x-axis at a single point. In a situation of equal roots, the vertex of the parabola lies precisely on the x-axis.

It's crucial to understand this concept to solve problems where equal roots must be determined or verified for a given equation. Equal roots symbolize the uniqueness of solutions and offer a direct way to connect the graphical representation with its algebraic form.

Discriminant

The discriminant of a quadratic equation is an essential tool in determining the nature of its roots. Given a standard quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is calculated using the formula \( D = b^2 - 4ac \). The value of the discriminant tells us not only if the roots are real or complex, but also if they are equal or distinct.

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), the quadratic equation has two equal real roots. It's at this point that the vertex of the parabola precisely touches the x-axis.
• If \( D < 0 \), the roots are complex and occur in a conjugate pair, meaning they don't intersect the x-axis at all.

This value provides a simple yet powerful insight into the behavior of a quadratic equation. It's a gateway to understanding how an equation behaves and is crucial when solving equations accurately by determining the right type of solution.

Roots of Quadratic Equation

Understanding the roots of a quadratic equation is fundamental to mastering quadratic solutions. A quadratic equation, expressed as \( ax^2 + bx + c = 0 \), can have two solutions, commonly referred to as roots. These roots are the x-values where the quadratic function intersects the x-axis. The solutions, or roots, can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula derives directly from manipulating the standard form of the quadratic equation through completing the square or using another algebraic method.

The "±" symbol highlights that there are generally two potential roots: one for plus and one for minus.

• If the value under the square root (the discriminant) is positive, both roots are distinct and real.
• If it is zero, both roots are equal, and the quadratic discriminant signifies this condition.
• However, if it's negative, the roots are complex and no real intercept occurs on the x-axis.

These roots are not just numbers; they represent crucial points where a quadratic equation lets us intersect with real-world values, such as optimization problems, physics trajectories, and so much more. Knowing how to effectively find roots using this knowledge can illuminate complex polynomial behaviors and solutions.