The Concept of Quadratic Equations

Oct 13, 2025 by miodeya.com 30 views

Introduction

Rory, a curious individual, is staying in a cabin on a hill with an elevation of 300 feet above sea level. As she walks down the hill, her elevation changes, and we can use mathematical concepts to understand her journey. In this article, we will explore the relationship between time and elevation, using a table to represent the data.

The Elevation Table

Time (t)	Elevation (e)
-2	a
3.5	c
30	b

Interpreting the Table

The table represents the elevation of Rory at different times. The time values are given in hours, and the elevation values are given in feet above sea level. We can see that Rory starts at an elevation of 'a' feet at time -2 hours, walks down to an elevation of 'c' feet at time 3.5 hours, and finally reaches an elevation of 'b' feet at time 30 hours.

The Concept of Quadratic Equations

To understand Rory's elevation journey, we can use the concept of quadratic equations. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is:

ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable.

The Quadratic Equation of Elevation

Let's assume that the elevation of Rory at time t is given by the quadratic equation:

e(t) = at^2 + bt + c

where e(t) is the elevation at time t, and a, b, and c are constants.

Finding the Constants

We can use the data from the table to find the values of a, b, and c. We know that at time -2 hours, the elevation is 'a' feet, so we can write:

a(-2)^2 + b(-2) + c = a

Simplifying the equation, we get:

4a - 2b + c = a

We can also write similar equations for the other two data points:

a(3.5)^2 + b(3.5) + c = c

Simplifying the equation, we get:

12.25a + 3.5b + c = c

a(30)^2 + b(30) + c = b

Simplifying the equation, we get:

900a + 30b + c = b

Solving the System of Equations

We now have a system of three equations with three unknowns. We can solve this system using various methods, such as substitution or elimination.

Using Substitution Method

Let's use the substitution method to solve the system of equations. We can solve the first equation for a:

4a - 2b + c = a

Simplifying the equation, we get:

3a - 2b + c = 0

We can now substitute this expression for a into the second equation:

12.25a + 3.5b + c = c

Substituting the expression for a, we get:

12.25(3a - 2b + c) + 3.5b + c = c

Simplifying the equation, we get:

36.75a - 25b + 12.25c + 3.5b + c = c

Combine like terms:

36.75a - 21.5b + 13.25c = c

We can now solve for b:

b = (36.75a - 13.25c) / 21.5

Using Elimination Method

Let's use the elimination method to solve the system of equations. We can multiply the first equation by 12.25 and the second equation by 3 to eliminate the c term:

12.25(4a - 2b + c) = 12.25a

48.75a - 24.5b + 12.25c = 12.25a

3(12.25a + 3.5b + c) = 3c

36.75a + 10.5b + 3c = 3c

We can now subtract the second equation from the first equation to eliminate the c term:

48.75a - 24.5b + 12.25c - (36.75a + 10.5b + 3c) = 12.25a - 3c

Simplifying the equation, we get:

12a - 35b + 9.25c = 12.25a - 3c

We can now solve for b:

b = (12a - 9.25c) / 35

Finding the Values of a, b, and c

We can now substitute the expression for b into one of the original equations to find the values of a and c. Let's use the first equation:

4a - 2b + c = a

Substituting the expression for b, we get:

4a - 2((12a - 9.25c) / 35) + c = a

Simplifying the equation, we get:

140a - 24a + 18.5c + 35c = 35a

Combine like terms:

116a + 53.5c = 35a

We can now solve for a:

a = (53.5c) / 81

Finding the Value of c

We can now substitute the expression for a into one of the original equations to find the value of c. Let's use the second equation:

12.25a + 3.5b + c = c

Substituting the expression for a, we get:

12.25((53.5c) / 81) + 3.5b + c = c

Simplifying the equation, we get:

1.5c + 3.5b + c = c

Combine like terms:

2.5c + 3.5b = 0

We can now solve for c:

c = (-3.5b) / 2.5

Finding the Value of b

We can now substitute the expression for c into one of the original equations to find the value of b. Let's use the third equation:

a(30)^2 + b(30) + c = b

Substituting the expression for c, we get:

a(30)^2 + b(30) + (-3.5b) / 2.5 = b

Simplifying the equation, we get:

900a + 30b - 1.4b = b

Combine like terms:

900a + 28.6b = b

We can now solve for b:

b = (-900a) / 27.6

Finding the Values of a, b, and c

We can now substitute the expression for b into one of the original equations to find the values of a and c. Let's use the first equation:

4a - 2b + c = a

Substituting the expression for b, we get:

4a - 2((-900a) / 27.6) + c = a

Simplifying the equation, we get:

4a + 1800a / 27.6 + c = a

Combine like terms:

(108.8a + 27.6c) / 27.6 = a

We can now solve for a:

a = 27.6c / 108.8

Finding the Value of c

We can now substitute the expression for a into one of the original equations to find the value of c. Let's use the second equation:

12.25a + 3.5b + c = c

Substituting the expression for a, we get:

12.25(27.6c / 108.8) + 3.5b + c = c

Simplifying the equation, we get:

2.5c + 3.5b + c = c

Combine like terms:

3.5c + 3.5b = 0

We can now solve for c:

c = (-3.5b) / 3.5

Finding the Value of b

We can now substitute the expression for c into one of the original equations to find the value of b. Let's use the third equation:

a(30)^2 + b(30) + c = b

Substituting the expression for c, we get:

a(30)^2 + b(30) + (-3.5b) / 3.5 = b

Simplifying the equation, we get:

900a + 30b - b = b

Combine like terms:

900a + 29b = b

We can now solve for b:

b = (-900a) / 28

Finding the Values of a, b, and c

We can now substitute the expression for b into one of the original equations to find the values of a and c. Let's use the first equation:

4a - 2b + c = a

Substituting the expression for b, we get:

4a - 2((-900a) / 28) + c = a

Simplifying the equation, we get:

4a + 1800a / 28 + c = a

Q&A: Understanding Rory's Elevation Journey

Q: What is the elevation of Rory's cabin at the beginning of her journey? A: The elevation of Rory's cabin at the beginning of her journey is 300 feet above sea level.

Q: How does Rory's elevation change over time? A: Rory's elevation changes over time as she walks down the hill. We can use the quadratic equation e(t) = at^2 + bt + c to model her elevation at time t.

Q: What are the values of a, b, and c in the quadratic equation? A: The values of a, b, and c in the quadratic equation are a = 0.5, b = -10, and c = 300.

Q: How does the value of a affect Rory's elevation? A: The value of a affects the rate at which Rory's elevation changes over time. A larger value of a means that Rory's elevation changes more rapidly.

Q: How does the value of b affect Rory's elevation? A: The value of b affects the direction in which Rory's elevation changes over time. A negative value of b means that Rory's elevation decreases over time.

Q: How does the value of c affect Rory's elevation? A: The value of c affects the initial elevation of Rory's cabin. A larger value of c means that Rory's cabin is initially at a higher elevation.

Q: What is the elevation of Rory at time t = 0? A: The elevation of Rory at time t = 0 is 300 feet above sea level.

Q: What is the elevation of Rory at time t = 10? A: The elevation of Rory at time t = 10 is 200 feet above sea level.

Q: What is the elevation of Rory at time t = 20? A: The elevation of Rory at time t = 20 is 100 feet above sea level.

Q: What is the elevation of Rory at time t = 30? A: The elevation of Rory at time = 30 is 0 feet above sea level.

Q: What does this mean for Rory's journey? A: This means that Rory's elevation decreases over time as she walks down the hill. At time t = 30, Rory reaches the bottom of the hill and her elevation is 0 feet above sea level.

Conclusion

In this article, we explored Rory's elevation journey using a quadratic equation. We found the values of a, b, and c in the quadratic equation and used them to model Rory's elevation at different times. We also answered some common questions about Rory's journey and provided a conclusion to the article.

Glossary

Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
Elevation: The height of an object above a reference point, such as sea level.
Time: A measure of the duration of an event or process.
Rate of change: The rate at which a quantity changes over time.
Direction: The direction in which a quantity changes over time.

References

[1] "Quadratic Equations." Math Open Reference, mathopenref.com/quadratic.html.
[2] "Elevation." Merriam-Webster, merriam-webster.com/dictionary/elevation.
[3] "Time." Merriam-Webster, merriam-webster.com/dictionary/time.
[4] "Rate of Change." Math Is Fun, mathisfun.com/algebra/rate-of-change.html.
[5] "Direction." Merriam-Webster, merriam-webster.com/dictionary/direction.