Solving the Equation ( x^2 - 5x 50 ): A Comprehensive Guide

A common algebraic problem involves solving quadratic equations, such as x2 - 5x 50. Let's explore the detailed steps in solving this fascinating equation and understand the underlying principles of quadratic equations and their solutions.

Understanding Quadratic Equations

A quadratic equation is of the structure ax2 bx c 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation can be found through a variety of methods including factoring, completing the square, and using the quadratic formula. In this guide, we will focus on the quadratic formula method, which is applicable to all quadratic equations.

Solving the Equation ( x^2 - 5x - 50 0 )

The given equation is ( x^2 - 5x - 50 0 ). To find the solutions, we will use the quadratic formula: [ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ] However, let's start by rewriting the equation in standard form. Adding 50 to both sides, we get:

[ x^2 - 5x  50 ]

Next, we move all terms to one side to set the equation equal to zero:

[ x^2 - 5x - 50  0 ]

Now, let's identify the coefficients: a 1, b -5, c -50. Applying the quadratic formula:

[ x frac{-(-5) pm sqrt{(-5)^2 - 4(1)(-50)}}{2(1)} ]

The discriminant is: [ (-5)^2 - 4(1)(-50) 25 200 225 ]. Therefore, we have:

Total    [ x  frac{5 pm sqrt{225}}{2} ]

Further simplifying, we get two potential solutions:

[ x  frac{5   15}{2}  frac{20}{2}  10 ]
[ x  frac{5 - 15}{2}  frac{-10}{2}  -5 ]
Thus, the solutions for x are:

x  10
x  -5
Verification of Solutions
To ensure the accuracy of the solutions, we can substitute them back into the original equation.
For x  10:
[ 10^2 - 5(10)  100 - 50  50 ]
For x  -5:
[ (-5)^2 - 5(-5)  25   25  50 ]
Both solutions satisfy the original equation, confirming their correctness.
Alternative Methods of Solving
While the quadratic formula provides a reliable method, there are other ways to solve similar equations. For instance, the equation can be factored:
[ x^2 - 5x - 50  (x   5)(x - 10)  0 ]
Setting each factor to zero:
x   5  0 ? x  -5
x - 10  0 ? x  10
This confirms our earlier solutions.
Conclusion
The solution to the equation (x^2 - 5x - 50  0) is x  -5 and x  10. These solutions were derived using both the quadratic formula and factoring methods, ensuring the solutions are both accurate and comprehensive.
Understanding the steps and methods used in solving such equations is crucial for mastering higher-level mathematics. By practicing and understanding the principles, students can tackle more complex problems with confidence.
Feel free to explore more equations and enhance your algebra skills. Happy learning!