Introduction to Quadratic Equations and Their Roots
In mathematics, a quadratic equation is an equation involving a variable (let's denote it by (x)) raised to the second power, typically written in the form of (ax^2 bx c 0). The roots of such an equation can be found using various methods, including the quadratic formula, factoring, and completing the square. In this article, we explore how to find the sum of roots for a specific quadratic equation, (x^2 - 3x 2 0).
Understanding the Quadratic Equation
Consider the quadratic equation (x^2 - 3x 2 0). This equation is in the standard form (ax^2 bx c 0), where (a 1), (b -3), and (c 2).
The Sum of Roots for Quadratic Equations
The sum of the roots of a quadratic equation (ax^2 bx c 0) can be determined using the formula (-frac{b}{a}). Applying this formula to the equation (x^2 - 3x 2 0), we find:
Sum of roots -frac{b}{a} -frac{-3}{1} 3
This means that the sum of the roots of the equation (x^2 - 3x 2 0) is 3.
Factoring the Quadratic Equation
Another method to solve the quadratic equation is by factoring. We start by finding the roots using the equation:
x^2 - 3x 2 0
We can factorize the quadratic expression:
(x - 1)(x - 2) 0
This implies that the roots of the equation are (x 1) and (x 2).
The sum of the roots is:
1 2 3
Conclusion
By using the sum of roots formula and factoring, we have verified that the sum of the roots of the quadratic equation (x^2 - 3x 2 0) is indeed 3. This conclusion applies to any quadratic equation of the form (ax^2 bx c 0), as the sum of the roots is always (-frac{b}{a}).
Additional Insights
Understanding the sum of the roots in a quadratic equation can be useful in various mathematical contexts, such as analyzing the behavior of quadratic functions, solving real-world problems involving quadratic relationships, and simplifying the process of finding roots through algebraic manipulation.
In conclusion, the sum of the roots of the equation (x^2 - 3x 2 0) is 3, as verified through both the formula and factorization methods.