Problem 39 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the roots of the equation \(x^{2}-p x+q=0\) differ by unity, then prove that \(p^{2}=4 q+1\).

Step-by-Step Solution

Answer

If the roots of the equation \(x^2 - px + q = 0\) differ by unity, we can denote the roots as \(a\) and \(b\), and substitute the roots in the sum and product equations: \(a + b = p\) and \(ab = q\). By solving for the linear term in terms of the roots and substituting into the product term, we find the relationship between \(p\) and \(q\). Simplifying the equation ultimately leads to the proof that \(p^2 = 4q + 1\).

Step 1: Identify the roots of the equation

Let's say the roots of the quadratic equation \(x^2 - px + q = 0\) are \(a\) and \(b\). Given that the roots differ by unity, we have either: 1. \(a = b + 1\) or 2. \(b = a + 1\)

Step 2: Use the properties of the roots

We know that the sum of the roots of a quadratic equation is equal to the ratio of the coefficients of the linear term to the coefficient of the quadratic term, and the product of the roots is equal to the ratio of the constants to the coefficient of the quadratic term. In our case, we have: Sum of the roots: \(a + b = p\) Product of the roots: \(ab = q\)

Step 3: Substitute the roots with the given condition

Using the first given condition, let's substitute the roots in the sum and product equations above: Sum of the roots: \((b + 1) + b = p\) Product of the roots: \((b + 1) * b = q\) Let's solve the sum of the roots equation for \(b\) and use it to find the relationship between \(p\) and \(q\).

Step 4: Solve for the linear term

Solve for the linear term in terms of the roots: \(2b + 1 = p \Rightarrow b = \frac{p - 1}{2}\)

Step 5: Substitute the linear term into the product term

Now, substitute \(b = \frac{p - 1}{2}\) into the equation of the product of the roots: \(\left(\frac{p - 1}{2} + 1\right) * \frac{p - 1}{2} = q\)

Step 6: Solve for the quadratic term

Simplify the equation to find the relationship between \(p\) and \(q\): \(\left(\frac{p + 1}{2}\right) * \frac{p - 1}{2} = q\) \( \frac{(p + 1)(p - 1)}{4} = q\) \( \frac{p^2 - 1}{4} = q\)

Step 7: Prove the equation

Now, prove the equation \(p^2 = 4q + 1\): \(p^2 - 4q = 1 \Rightarrow p^2 = 4q + 1\) Hence, we have proved that if the roots of the equation \(x^2 - px + q = 0\) differ by unity, then \(p^2 = 4q + 1\).

Key Concepts

Roots of a Quadratic Equation

Understanding the roots of a quadratic equation is fundamental in algebra. A quadratic equation in the form of \(ax^2 + bx + c = 0\) has two solutions or roots, which could be real or complex numbers. These roots are the values of \(x\) that make the equation true.

For the given problem where the quadratic equation is \(x^2 - px + q = 0\), we symbolize the roots as \(a\) and \(b\). The condition that these roots differ by unity, which means one unit, tells us that they are consecutive numbers, such as 3 and 4, or can be represented as \(a = b + 1\) or \(b = a + 1\). Solving for the roots explicitly isn't necessary in our scenario; instead, we exploit the relationship between the roots to find the desired proof.

Sum and Product of Roots

The sum and product of the roots of a quadratic equation reveal a powerful connection with the coefficients of the equation. They are based on the formulas: sum of the roots, \(a + b\), being \(-\frac{b}{a}\) when the quadratic equation is given by \(ax^2 + bx + c = 0\) and product of the roots, \(ab\), being \(\frac{c}{a}\).

Applying these to the given quadratic equation \(x^2 - px + q = 0\), where \(a = 1\), \(b = -p\), and \(c = q\), the sum of the roots equation becomes \(a + b = p\) and the product \(ab = q\). This direct relationship allows us to derive expressions involving \(p\) and \(q\) which can be manipulated algebraically to reach the solution without actually finding the roots themselves.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables they contain. These are particularly useful for simplifying expressions and solving equations. In the context of quadratic equations, the most relevant identities are those related to the squares and products of binomials, such as \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)(a + b) = a^2 - b^2\).

In the final steps of our solution, we see the latter identity being applied when we encounter \(\frac{(p + 1)(p - 1)}{4} = q\). This simplifies to \(\frac{p^2 - 1}{4} = q\), allowing us to directly compare this form to the given equation \(p^2 = 4q + 1\). Through algebraic identities, we elegantly bridge between the sum and product of the roots to the desired relationship, reinforcing not only the power of these identities but also their critical role in understanding and solving quadratic equations.