Solving x^2 1 25 0: A Comprehensive Guide to Quadratic Equations and Factoring
Quadratic equations are integral to algebra, and understanding how to solve them is key for many applications in mathematics, engineering, and physics. This article provides a detailed look at solving the specific problem of x2 1 25 0 using the method of factoring. We will explore the concept, step-by-step solution, and the significance of the roots.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, and is generally expressed in the form ax2 bx c 0. The solutions of a quadratic equation are known as roots or zeros of the equation. These roots represent the x-coordinates of the points where the graph of the function intersects the x-axis.
Solving x2 1 25 0 by Factoring
To solve the equation x2 1 25 0, we start by factoring the left-hand side of the equation.
Step 1: Identify the Coefficients
The coefficients are a 1, b 10, and c 25. The aim is to express the equation as (x - a)(x - b) 0 where ab 25 and a b 10.
By factoring, we find that (x 5)(x 5) 0 or (x 5)2 0.
Step 2: Apply the Zero Product Rule
According to the zero product rule, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we have:
x 5 0
Solving for x gives:
x -5
Step 3: Verification
To verify the solution, substitute x -5 into the original equation:
x2 1 25 0
Replacing x with -5:
(-5)2 10(-5) 25 25 - 50 25 0
This confirms that x -5 is indeed a solution.
The Significance of the Roots
The root of the equation, x -5, represents the x-coordinate of the point where the parabola defined by the function y x2 1 25 intersects the x-axis. The graph of this function is an up-opening parabola with a vertex at the point (-5, 0).
Conclusion
In conclusion, the equation x2 1 25 0 can be solved by factoring it into (x 5)2 0. The solution is x -5, which is the root of the equation. Understanding how to solve such equations is fundamental to many areas of mathematics and its applications.