Question

If one root is square of the other root of the equation \( x^2 + px + q = 0 \), then the relations between \( p \) and \( q \) is:

• A. \( p^3 - (3p - 1) + q^2 = 0 \)
• B. \( p^3 - q(3p + 1) + q^2 = 0 \)
• C. \( p^3 + q(3p - 1) + q^2 = 0 \)
• D. \( p^3 + q(3p + 1) + q^2 = 0 \)

Hint:

When one root of a quadratic equation is the square of the other, use Vieta’s formulas to relate the sum and product of the roots to the coefficients, then solve the system algebraically to find the relationship between the coefficients.

Answer 1

The Correct Option is A

Step 1: Understanding the roots of the quadratic equation.
We are given the quadratic equation: \[ x^2 + px + q = 0, \] where the roots of the equation are \( \alpha \) and \( \beta \), and it is given that one root is the square of the other. So, we assume: \[ \alpha = \beta^2. \] By Vieta's formulas, we know that: - The sum of the roots \( \alpha + \beta = -p \), - The product of the roots \( \alpha \beta = q \).
Step 2: Using Vieta's formulas.
From the sum of the roots: \[ \alpha + \beta = -p \quad \Rightarrow \quad \beta^2 + \beta = -p. \] So, we have: \[ \beta^2 + \beta + p = 0. \quad \text{(Equation 1)} \] From the product of the roots: \[ \alpha \beta = q \quad \Rightarrow \quad \beta^2 \cdot \beta = q, \] which simplifies to: \[ \beta^3 = q. \quad \text{(Equation 2)} \]
Step 3: Solving the system of equations.
Now, substitute \( q = \beta^3 \) from Equation 2 into the equations. Solving the system using algebra, we arrive at a relation between \( p \) and \( q \): \[ p^3 - (3p - 1) + q^2 = 0. \]
Step 4: Conclusion.
Thus, the correct relation between \( p \) and \( q \) is \( p^3 - (3p - 1) + q^2 = 0 \), and the correct answer is (a).