How to Write a Quadratic Equation with Given Roots: A Comprehensive Guide

When tasked with finding a quadratic equation with specific roots, such as -2 and 3, the process can be straightforward with the help of some basic algebra and Vieta's formulas. This guide will walk you through the steps to derive the quadratic equation from the roots, as well as the mathematical reasoning behind it.

Understanding Quadratic Equations with Given Roots

A quadratic equation is a polynomial equation of the second degree. Typically, it is written in the form:

ax2 bx c 0

The roots of a quadratic equation are the values of x that satisfy the equation. If we are given the roots, we can easily construct the quadratic equation. For roots -2 and 3, let's see how we can arrive at the equation.

Constructing the Quadratic Equation from Roots

Direct Method Using Roots

Given the roots -2 and 3, we can start with the factored form of the quadratic equation:

x - (-2)(x - 3) 0

Simplifying this, we get:

x 2(x - 3) 0

Further simplification leads to:

x2 - 3x 2x - 6 0

This simplifies to:

x2 - x - 6 0

Using Vieta's Formulas for Verification

Vieta's formulas provide a way to relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation ax2 bx c 0 with roots a and b, Vieta's formulas state:

-b/a a b c/a a * b

Using the given roots -2 and 3:

-b/a -2 3 1 c/a -2 * 3 -6

Assuming a 1, we have:

b -1 c -6

This means the quadratic equation is:

x2 - x - 6 0

General Solution with Leading Coefficient

For a quadratic equation with leading coefficient a ≠ 1, the equation can be scaled as:

a(x2 - x - 6) 0

This retains the same roots, but scales the entire equation. Therefore, for any non-zero value of a, the equation a(x2 - x - 6) 0 is valid.

Step-by-Step Construction of the Quadratic Equation

Create a quadratic 'guide': x - _x - _ 0 Fill in the blanks with each root: x - (-2)x - 3 0 Simplify: x 2x - 3 0 Optional: Simplify further: x2 - x - 6 0

Conclusion

In conclusion, the quadratic equation with roots -2 and 3 can be derived in multiple ways, but the most common approach after factoring is to expand and simplify. The equation we obtain is:

x2 - x - 6 0

This equation not only has the specified roots but can also be scaled by any non-zero constant a. Understanding the process and the underlying mathematical principles can make similar problems easier to solve.