Solving for Speed in Quadratic Equations
In this article, we will explore how to solve a problem related to speed, distance, and time using quadratic equations. This method is commonly used in various mathematical and real-world applications, such as transportation and travel analysis. Let's dive into the details and break down the problem step by step.
Problem Statement
Suppose a person travels a distance of 12 kilometers in two different scenarios. In the first scenario, the person travels at a speed of V1 km/hr for a time of T1 hours. In the second scenario, the person travels the same distance but at a higher speed of V2 km/hr, taking a time of T2 hours. The relationship between T2 and T1 is given by T2 T1 - 8/60 (since 8 minutes is 8/60 hours).
Solving the Quadratic Equation
The basic formula that governs the relationship between time, speed, and distance is:
TV D
Where T is time in hours, V is velocity in km/hr, and D is distance in km. Therefore, for the first scenario, we have:
T1V1 12
For the second scenario:
T2V2 12
Given that T2 T1 - 8/60 and V2 3V1, we can substitute these into the second equation:
(T1 - 8/60)V2 12
Substituting V2 with 3V1 gives:
(T1 - 8/60)3V1 12
Normalization and Simplification
To further simplify, we can use the equation TV T - 8/60V3. We can also express T in terms of 12/V, hence:
V1 12/T
Substituting V1 with 12/T and V2 with 3(12/T), we get:
(T - 8/60)(12/T) 12
Multiplying both sides by T results in:
12 - 96/60 12
Further simplifies to:
12 - 16/10 12
Which is a quadratic equation in the form:
s2 - 3s - 270 0
Determining the Number of Real Roots
In order to determine the number of real roots of a quadratic equation, we can use the discriminant, where the discriminant D b2 - 4ac. Here, a 1, b -3, and c -270. Substituting these values, we get:
D (-3)2 - 4(1)(-270) 9 1080 1089
Since D > 0, the quadratic equation has two distinct real roots.
Conclusion
In conclusion, the quadratic equation derived from the problem statement is s2 - 3s - 270 0. This equation can be solved using standard algebraic methods, resulting in two real roots. The discriminant confirms that there are indeed two real roots, which means that the problem has two valid solutions for the speed at which the person was traveling in the second scenario.
References
This problem is taken from various resources on quadratic equations and their applications in real-world scenarios. The derivation and solution are based on the standard methods used to solve such problems.
Keywords
quadratic equation, speed, distance, problem solving
Tags
#quadratieequation #distance #speed #problem_solving #algebra