Solving Quadratic Equations: A Case Study
Quadratic equations often present complex yet solvable problems in algebra. This article delves into a specific example, illustrating quadratic equation solving techniques while highlighting the importance of mathematical reasoning in finding clear solutions. Let's explore how solving equations can be applied to real-world problem scenarios.
The Equation in Question
The given equation is: x^2 17x - 70. This problem can be simplified and solved to find the value of x.
Simplification and Solution Steps
First, we move all terms to one side to form a standard quadratic equation:
x^2 - 17x 70 0
Factoring the Equation
The equation can be factored into two binomials, as follows:
(x - 7)(x - 10) 0
This factorization means that x - 7 0 or x - 10 0. Solving these two simple equations gives us:
x 7 or x 10
Proof of Solution
To verify the solution, let's substitute each value back into the original equation:
For x 7：
7^2 17(7) - 70
196 119 - 70
196 49
For x 10：
10^2 17(10) - 70
100 170 - 70
100 100
Both values satisfy the original equation, confirming that x 7 and x 10 are correct solutions.
Understanding the Roots
The two roots, 7 and 10, represent the points where the quadratic equation touches the x-axis on a graph. This understanding is crucial for broader applications in areas like physics, economics, and engineering, where quadratic equations are often used to model various phenomena.
Conclusion
Quadratic equations, with their characteristic form x^2 17x - 70, can be solved by various methods, including factoring. This example showcases the importance of algebraic skills in resolving seemingly complex problems. Whether you're dealing with real-world applications or theoretical explorations, mastering these techniques provides a strong foundation in mathematics.
References and Further Reading
For further exploration, you may consult any standard algebra textbook or relevant online resources which provide in-depth coverage of quadratic equations and their applications.