How to Write a Quadratic Equation Given its Roots
Finding a quadratic equation when you know its roots is a fundamental concept in algebra. This skill is crucial for various mathematical applications and problem-solving. This guide will walk you through the process, explaining the underlying principles and providing examples.
Understanding Quadratic Equations and Roots
A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. 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 representing the quadratic function.
The Relationship Between Roots and Coefficients
The key to constructing a quadratic equation from its roots lies in understanding Vieta's formulas. These formulas relate the roots of a quadratic equation to its coefficients:
- Sum of Roots: The sum of the roots (α + β) is equal to -b/a.
- Product of Roots: The product of the roots (αβ) is equal to c/a.
Steps to Write a Quadratic Equation Given the Roots
Let's assume the roots of the quadratic equation are α and β. Here's a step-by-step guide:
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Start with the factored form: A quadratic equation can be written in factored form as a(x - α)(x - β) = 0, where 'a' is a constant (it can be any non-zero real number). This form directly uses the roots.
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Expand the equation: Multiply out the factored form to obtain the standard quadratic equation form: ax² + bx + c = 0. This step involves expanding the expression (x - α)(x - β).
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Simplify (if necessary): If 'a' is not 1, you might choose to simplify by dividing the entire equation by 'a', to obtain a simpler equation with integer coefficients. However, this step is optional.
Examples
Let's illustrate this with a few examples:
Example 1: Roots are 2 and 3
- Factored form: a(x - 2)(x - 3) = 0
- Expansion: a(x² - 5x + 6) = 0
- Simplified (with a = 1): x² - 5x + 6 = 0
Therefore, the quadratic equation is x² - 5x + 6 = 0.
Example 2: Roots are -1 and 4
- Factored form: a(x + 1)(x - 4) = 0
- Expansion: a(x² - 3x - 4) = 0
- Simplified (with a = 1): x² - 3x - 4 = 0
Therefore, the quadratic equation is x² - 3x - 4 = 0.
Example 3: Roots are ½ and -3
- Factored form: a(x - ½)(x + 3) = 0
- Expansion: a(x² + (5/2)x - (3/2)) = 0
- Simplified (with a = 2): 2x² +5x -3 = 0
Therefore, the quadratic equation is 2x² + 5x - 3 = 0.
Important Considerations
- The constant 'a': You can choose any non-zero value for 'a'. Selecting 'a = 1' often simplifies the equation.
- Complex Roots: The method works equally well for quadratic equations with complex roots. The process of expanding the factored form remains the same.
- Repeated Roots: If a quadratic equation has repeated roots (e.g., α = β), the factored form simplifies to a(x - α)² = 0.
By following these steps, you can confidently construct a quadratic equation given its roots, regardless of the nature of those roots. This understanding is key to mastering quadratic equations and their applications in various mathematical contexts.
