Write A Quadratic Equation With The Given Roots

Writing Quadratic Equations with Given Roots: A Comprehensive Guide

Finding the quadratic equation when you know its roots is a fundamental concept in algebra. This guide provides a comprehensive walkthrough, covering various scenarios and offering practical examples to solidify your understanding. We'll explore the standard form, factored form, and the relationship between roots and coefficients, ensuring you can confidently tackle any problem. This will include discussions on:

Understanding Quadratic Equations and Their Roots

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. It's generally expressed in the standard form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The roots (or solutions) of a quadratic equation are the values of x that satisfy the equation. These roots represent the x-intercepts of the parabola represented by the quadratic function.

There are several ways to find the roots, including:

• Factoring: Expressing the quadratic as a product of two linear factors.
• Quadratic Formula: Using the formula x = (-b ± √(b² - 4ac)) / 2a
• Completing the Square: Manipulating the equation to create a perfect square trinomial.

From Roots to Equation: The Core Concept

The key to constructing a quadratic equation from its roots lies in understanding the factor theorem. This theorem states that if r is a root of a polynomial equation, then (x - r) is a factor of the polynomial.

Let's say we have two roots, α and β. According to the factor theorem, (x - α) and (x - β) are the factors of the quadratic equation. Therefore, the quadratic equation can be written in factored form as:

(x - α)(x - β) = 0

Expanding this equation will give us the standard form of the quadratic equation.

Constructing Quadratic Equations: Step-by-Step Examples

Let's walk through several examples to illustrate different scenarios:

Example 1: Real and Distinct Roots

Problem: Find the quadratic equation whose roots are 2 and -3.

Solution:

1. Identify the roots: α = 2 and β = -3

2. Write the factored form: (x - 2)(x - (-3)) = 0

3. Simplify and expand: (x - 2)(x + 3) = 0

4. Expand to standard form: x² + 3x - 2x - 6 = 0

5. Final Equation: x² + x - 6 = 0

Example 2: Real and Equal Roots (Repeated Roots)

Problem: Find the quadratic equation with a repeated root of 5.

Solution:

1. Identify the roots: α = 5 and β = 5

2. Write the factored form: (x - 5)(x - 5) = 0

3. Simplify and expand: (x - 5)² = 0

4. Expand to standard form: x² - 10x + 25 = 0

Example 3: Complex Roots (Conjugate Pairs)

Complex roots of quadratic equations always come in conjugate pairs. A conjugate pair means that if (a + bi) is a root, then (a - bi) is also a root, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1).

Problem: Find the quadratic equation whose roots are 2 + i and 2 - i.

Solution:

1. Identify the roots: α = 2 + i and β = 2 - i

2. Write the factored form: (x - (2 + i))(x - (2 - i)) = 0

3. Expand carefully: This requires careful expansion, paying close attention to the complex numbers. We can use the difference of squares:

   [(x - 2) - i][(x - 2) + i] = 0

   (x - 2)² - (i)² = 0

   (Remember that i² = -1)

4. Simplify and expand: (x - 2)² + 1 = 0

5. Expand to standard form: x² - 4x + 4 + 1 = 0

6. Final Equation: x² - 4x + 5 = 0

Example 4: Using the Sum and Product of Roots

The sum and product of the roots of a quadratic equation ax² + bx + c = 0 have a direct relationship with the coefficients:

• Sum of roots (α + β) = -b/a
• Product of roots (αβ) = c/a

We can use these relationships to construct the quadratic equation.

Problem: Find the quadratic equation whose roots have a sum of 7 and a product of 12.

Solution:

1. Identify the sum and product: α + β = 7 and αβ = 12

2. Assume a = 1 (for simplicity): This simplifies the relationships to: α + β = -b and αβ = c

3. Determine b and c: b = -7 and c = 12

4. Write the equation: x² - 7x + 12 = 0

Handling Irrational and Fractional Roots

The principles remain the same when dealing with irrational or fractional roots.

Problem: Find the quadratic equation whose roots are (1 + √2) and (1 - √2).

Solution:

1. Identify the roots: α = 1 + √2 and β = 1 - √2

2. Write the factored form: [x - (1 + √2)][x - (1 - √2)] = 0

3. Expand carefully: (x - 1 - √2)(x - 1 + √2) = 0

   [(x - 1) - √2][(x - 1) + √2] = 0

   (x - 1)² - (√2)² = 0

4. Simplify and expand: x² - 2x + 1 - 2 = 0

5. Final Equation: x² - 2x - 1 = 0

Advanced Considerations and Applications

This core concept extends to higher-degree polynomial equations. While the factoring method becomes more complex for cubic or higher-degree polynomials, the fundamental principle of using factors derived from roots remains valid.

Understanding the relationship between roots and coefficients is crucial in various applications, including:

• Curve Fitting: Determining the equation of a parabola that passes through specific points.
• Signal Processing: Analyzing and designing filters.
• Physics and Engineering: Modeling parabolic trajectories and other phenomena.

Conclusion

Constructing a quadratic equation from its roots is a straightforward yet powerful technique with far-reaching applications. By understanding the factor theorem and the relationship between roots and coefficients, you can confidently tackle a wide range of problems involving quadratic equations. Remember to pay close attention to the signs, particularly when dealing with negative roots and complex numbers. Mastering this skill is essential for anyone seeking a strong foundation in algebra and its various applications. Practice consistently, working through diverse examples to build your proficiency and confidence.