Quadratic Equations | Properties of Roots (Sum & Product, Discriminant)

Learning Objectives

By the end of this article, you should be able to:

• Derive and apply the formulas for the sum and product of roots of a quadratic equation.
• Construct a quadratic equation given its roots.
• Solve parameter problems using sum and product of roots.
• Analyze the signs of roots using discriminant, sum, and product.
• Evaluate symmetric expressions in terms of \( \alpha + \beta \) and \( \alpha\beta \).
• Apply all these skills to DSE-style questions.

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1. Introduction

The properties of roots of quadratic equations appear every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:

• Find the sum or product of roots without solving the equation
• Construct a quadratic equation from given roots
• Find unknown parameters using root properties
• Analyze the signs of roots
• Evaluate expressions involving roots

These are high-level skills that can help you secure top marks in the exam.

2. Sum and Product of Roots

For a quadratic equation \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \):

These formulas are derived from the factorization:

Worked Example

Question: Find the sum and product of roots of \( 2x^2 - 5x + 3 = 0 \).

3. Constructing a Quadratic Equation from Roots

If \( \alpha \) and \( \beta \) are roots, the quadratic equation is:

Worked Example

Question: Construct a quadratic equation with roots \( 3 \) and \( -2 \).

4. Symmetric Expressions

Symmetric expressions are expressions that remain the same if \( \alpha \) and \( \beta \) are swapped. They can be evaluated using \( \alpha + \beta \) and \( \alpha\beta \).

Common Formulas

• \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \)
• \( \alpha^2 - \beta^2 = (\alpha - \beta)(\alpha + \beta) \)
• \( \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) \)
• \( \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \)

Worked Example

Question: If \( \alpha \) and \( \beta \) are roots of \( x^2 - 4x + 2 = 0 \), find \( \alpha^2 + \beta^2 \).

5. Parameter Problems

Parameter problems involve finding unknown coefficients in a quadratic equation using the sum and product of roots.

Worked Example

Question: The roots of \( x^2 + kx + 6 = 0 \) are \( 2 \) and \( 3 \). Find \( k \).

6. Sign Analysis of Roots

Using the discriminant (\( \Delta \)), sum (\( S \)), and product (\( P \)) of roots, we can determine the signs of the roots without solving the equation.

Condition	Signs of Roots
\( \Delta \ge 0 \), \( S > 0 \), \( P > 0 \)	Both roots positive
\( \Delta \ge 0 \), \( S < 0 \), \( P > 0 \)	Both roots negative
\( \Delta \ge 0 \), \( P < 0 \)	One positive, one negative

Worked Example

Question: Determine the signs of the roots of \( x^2 - 5x + 6 = 0 \).

7. Worked Examples

Example 1: Sum and Product

Question: Find the sum and product of roots of \( 3x^2 + 7x - 2 = 0 \).

Example 2: Constructing from Roots

Question: Construct a quadratic equation with roots \( \frac{1}{2} \) and \( -3 \).

Example 3: Parameter Problem

Question: If one root of \( x^2 - kx + 12 = 0 \) is 4, find the other root and the value of \( k \).

8. DSE-Style Practice Questions

9. Solutions with Explanations

10. Exercise

Click the following link to have
 An Exercise on Quadratic Equations | Properties of Roots (Sum & Product, Discriminant)

Key Takeaways

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Summary Checklist for Revision

• Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of roots: \( \alpha\beta = \frac{c}{a} \)
• Construct equation: \( x^2 - (\alpha + \beta)x + \alpha\beta = 0 \)
• Symmetric expressions: \( \alpha^2 + \beta^2, \frac{1}{\alpha} + \frac{1}{\beta} \)
• Parameter problems: use sum/product to find unknowns
• Sign analysis: \( \Delta \ge 0 \), then \( S \) and \( P \)
• Check answers by substituting back