Solving the Quadratic Equation x2 - 2x - 3 0: A Comprehensive Guide
Quadratic Equations are fundamental in algebra and have a wide range of practical applications in fields such as engineering, physics, and even economics. A Quadratic Equation is any equation that can be written in the form (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0).
Solving x2 - 2x - 3 0 with the Quadratic Formula
The Quadratic Formula is a powerful method for solving quadratic equations and has the general form:
[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]
The quadratic equation we are solving is (x^2 - 2x - 3 0). Here, (a 1), (b -2), and (c -3).
Steps to Solve using the Quadratic Formula
Substitute the values of (a), (b), and (c) into the quadratic formula:
[x frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 1 cdot (-3)}}{2 cdot 1}]
Simplify the expression inside the square root:
[x frac{2 pm sqrt{4 12}}{2}]
[x frac{2 pm sqrt{16}}{2}]
Find the square root of 16:
[x frac{2 pm 4}{2}]
Calculate the two values for (x):
[x_1 frac{2 4}{2} 3]
[x_2 frac{2 - 4}{2} -1]
Solving x2 - 2x - 3 0 by Factoring
Another method for solving this quadratic equation is by factoring. To factor the equation (x^2 - 2x - 3 0), we need to find two numbers that multiply to (-3) and add to (-2).
The numbers that satisfy these criteria are (1) and (-3):
[x^2 - 2x - 3 (x 1)(x - 3) 0]
Set each factor equal to zero and solve for (x):
[x 1 0 implies x -1]
[x - 3 0 implies x 3]
Equations in Different Forms
The equation (x^2 - 2x - 3 0) can also be written in different forms:
Rearrangement:
[(x - 1)^2 - 4 0]
Factoring:
[(x - 1 - 2)(x - 1 2) 0]
Adding the Same Term:
[x^2 - x - x - 3 0]
[(x - 1)(x - 3) 0]
Conclusion
Both the Quadratic Formula and factoring are effective methods for solving the quadratic equation (x^2 - 2x - 3 0). In this case, the solutions are:
[x 3 text{ or } x -1]
Additional Resources
For more detailed information on solving quadratic equations, consider exploring advanced algebra courses or online resources dedicated to mathematics.