Word Problems Involving Quadratic Equation
For his livestock, a farmer wishes to build a rectangle pen. He wants to increase the pen size and has 100 meters of fencing material. What measurements should he employ?
We can begin by assuming that the pen’s length and width are represented by ‘x’ and ‘y’ meters, respectively, to answer this problem. We can infer that the opposite sides of the pen are of equal length because it is a rectangular object.
In this example, the sum of all the sides, twice the length plus twice the width, determines the pen’s perimeter. We can express the equation as 2x + 2y = 100 because the farmer has 100 meters of fencing material.
We now wish to maximize the pen’s surface area. Given that the area of a rectangle is equal to its length times its breadth, the area of the pen is equal to x * y.
We must translate the area into terms of a single variable to calculate the maximum area. We can solve the perimeter equation for a single variable and then replace it in the area equation because we already have the perimeter equation.
We can find y using the perimeter equation: 2x + 2y = 100. The equation is rewritten as y = (100 + 2x)/2 = 50 + x.
Now that y has been added, the area equation can be written as A = x * y = x * (50 + x).
To determine the largest area, we must determine the value of x that maximizes the quadratic function A = x * (50 x).
We can use calculus methods or a graph of the quadratic function to do this. We can find the crucial points by taking the area function’s derivative concerning x and setting it to zero. However, let’s attempt an algebraic solution:
A = x * (50 – x) = 50x – x2.
We locate the x-value that causes the derivative of A to equal zero to determine the maximum. We obtain dA/dx = 50 – 2x by differentiating A about x.
50 +minus 2x = 0 when the derivative is set to zero. We arrive at x = 25 after solving for x.
To maximize the size of the pen, the farmer should select a length of 25 meters and a width of (50 minus 25) = 25 meters.
We can confirm that the total amount of fencing material utilized is, in fact, 100 meters by inserting these values back into the perimeter equation: 100 meters equals 2 * 25 + 2 * 25 multiplied by 50.
So, using his 100 meters of fencing material, the farmer can construct a rectangle pen with a maximum area of 25 meters long by 25 meters wide.
How Do You Solve A Quadratic Equation With A Word Problem?
Mathematical equations involving a variable raised to the power of two are known as quadratic equations. These equations are frequently used in real-world contexts, including physics, engineering, and finance. You can use particular procedures to solve a word problem when presented with a quadratic equation.
Recognize The Issue
Understanding the material provided and the issue being posed in full is the first and most important stage in solving a quadratic equation in a word problem. Identify the quantities involved, their relationships, and the exact unknown value you need to find by carefully reading the problem. It can be useful to emphasize or highlight pertinent material to ensure no critical details are missed.
Define the variable that stands in for the unknown value once you fully grasp the issue. For instance, you can use “x” to denote the length of a square’s side if the problem asks for that information. After defining the variable, you can build the quadratic equation that describes the circumstances.
Formulate The Quadratic Equation
After the variable has been defined, construct a quadratic equation using the available data. In word problems, the equation frequently results from using an appropriate formula or comprehending the problem’s context. For instance, if the task is to determine the projectile’s maximum height, you can use the kinematic equation: h = -16t2 + vt + s, where “h” stands for height, “t” for time, “v” for initial velocity, and “s” for initial height.
Substituting the known values into the equation and simplifying it will result in a quadratic equation. Make sure that the equation is in the conventional form, ax2 + bx + c = 0, with “a,” “b,” and “c” being constants. Rearrange the terms such that zero is on one side of the equation and all the variables are on the other.
Apply The Quadratic Formula
The quadratic formula must then be used with the quadratic equation in standard form. According to the quadratic formula, the solutions for “x” in any quadratic equation of the type ax2 + bx + c = 0 are as follows:
x = [-b (b2 – 4ac)] / (2a]
The character of the solutions is determined by the discriminant (b2 + 4ac) in the formula. The equation has two unique real solutions if the discriminant is positive. A perfect square is the only genuine solution to the equation if the discriminant is zero. If the discriminant is negative, there are two complex conjugate solutions to the equation.
Solve For The Unknown Value
Substitute the values of “a,” “b,” and “c” from the quadratic equation produced in the previous step using the quadratic formula. Determine the discriminant’s worth by evaluating it. You can choose the quantity and type of solutions depending on the characteristics of the discriminant.
Evaluate both of the “x” solutions if the discriminant is positive. If it is zero, there is only one answer; there are complex number solutions if it is negative. In the quadratic equation, substitute the proper values, then find “x.”
How Do I Write An Equation From A Word Problem?
Word puzzles offer a realistic setting for using mathematical ideas and calculations. Creating an equation to describe the information provided and the unknown value when solving a word problem is one of the crucial tasks. In this post, we will walk you through the process of creating equations from word problems, breaking them down into clear steps.
Specify The Variables
The next step is to define the variables used in the equation once you fully grasp the issue. Variables often take the form of letters and denote unknowable values. You can build the equation that connects the variables by giving a letter to each unknown quantity.
For instance, you can use “C” to denote the overall cost and “n” to denote the number of products if the problem asks for the total cost of purchasing many goods. After defining the relationship between these variables, you can write an equation to represent them.
Identify The Operations And Relationships
After specifying the variables,
- Analyze the linkages and procedures outlined in the word problem.
- Look for terms or phrases denoting particular mathematics operations, such as addition, subtraction, multiplication, or division.
- Determine the trends or laws that control the relationships between the variables.
For instance, you can infer that the total cost “C” can be computed by multiplying the number of things “n” by the cost per item and then adding the sales tax if the problem indicates that the cost of each item is $10 and there is a 10% sales tax. The equation for this relationship is C = 10n + 0.10 (10n).
Simplify The Equation And Find Its Solution
When the equation is complete, combine like terms and simplify any necessary computations. Ensuring that the equation is in its most basic form, this step makes it simpler to find the unknown number.
Using the prior example as a guide, the equation C = 10n + 0.10 (10n0.10(10n)) may be broken down into C = 10.10n + 1n, which can then be broken down into C = 11.10n. When you know the quantity of things (“n”), you may use this equation to solve for the total cost (“C”) and see how the variables relate to one another.
Double-Check Your Equation
Before concluding, it is crucial to verify that the equation captures the nature of the given issue. In the equation, replace the unknown numbers with the known values after reviewing the original problem statement. Check to see if the equation is valid and correctly reflects the relationships in the word problem.
Making sure the equation is correct also makes it easier to see any mistakes or inconsistencies that might have happened throughout the formulation process. Review the steps and make any necessary modifications if the equation does not match the original issue. We will look at a few instances to demonstrate how word equations are created and solved.
Example: Time, Speed, and Distance
Finding the correlation between distance, speed, and time is one frequent use of word equations. Let’s imagine a situation where a car drives for a predetermined period at a fixed speed. The equation relates these two variables:
For instance, using the following equation, one may determine the distance traversed by a car traveling at 60 mph for two hours. Distance 60 mph times two hours equals 120 miles. In this case, the equation enables us to determine the car’s travel distance given its speed and time.
Example: Simple Interest
A simple interest calculation serves as yet another illustration of a word equation. Simple interest, which is frequently used in finance, is the extra money gained or paid on an initial sum over a specific period. Simple interest is calculated using the following formula:
For example, depositing $1,000 in a savings account with a 5% yearly interest rate The following formula can be used to determine the interest earned after two years:
Interest: $100 ($1,000 x 0.05 x 2)
The equation in this situation enables us to calculate the amount of interest earned by multiplying the principal, interest rate, and duration.
Example: A Rectangle’s Area
Another illustration of a word equation is the area of a rectangle. The following formula can be used to determine a rectangle’s area:
Area Equals Length x width
Let’s say we have a rectangle 4 meters wide and 6 meters long. The following equation can be used to get the rectangle’s area:
Six Meters X 4 Meters Is Equal To 24 Square Meters
In this case, using the rectangle’s specified length and width as inputs, the equation allows us to calculate the rectangle’s area.
Example: Newton’s Second Law of Motion
Physics also uses a lot of equations, like Newton’s second law of motion. The following equation, which describes the connection between force, mass, and acceleration,
Mas Plus Acceleration Equals Force
Think about an object that weighs 5 kilograms and accelerates by 10 meters per second squared. The following equation can be used to determine the force applied to the object:
50 Newtons equal 5 kilograms of force at 10 meters per second. Using the mass and acceleration of an item as inputs, we can use this equation to calculate the force acting on the object.
Word equations offer a way to convert issues from the actual world into mathematical terms. They help us address issues across disciplines and quantify correlations between variables. We can successfully solve difficult problems and get insights into the quantitative components of our daily experiences by comprehending and using word equations.
How do I set up a quadratic equation from a word problem?
To set up a quadratic equation, identify the unknown quantity and assign it a variable (usually denoted by “x”). Translate the given information into mathematical expressions or equations, and then form a quadratic equation by equating it to zero.
What are some common scenarios that can be represented by quadratic equations?
Quadratic equations often arise in problems involving projectile motion, area or perimeter calculations, maximizing or minimizing values, and problems related to time, distance, or speed.
How do I solve a quadratic equation in a word problem?
Once you’ve set up the quadratic equation, you can solve it using various methods such as factoring, completing the square, or using the quadratic formula. After finding the values of “x,” make sure they are meaningful in the context of the problem.
Can a quadratic equation have multiple solutions in a word problem?
Yes, a quadratic equation can have two real solutions, one real solution, or no real solutions, depending on the discriminant (b² – 4ac). The number of solutions corresponds to the number of distinct solutions in the word problem.
How can I check if the solutions of a quadratic equation are correct in a word problem?
Once you find the solutions for “x,” substitute them back into the original word problem and verify if the resulting values satisfy the given conditions. If the substituted values make the equation true, then the solutions are correct.
Are there any strategies to tackle word problems involving quadratic equations?
To solve word problems involving quadratic equations, follow these steps: a) Read the problem carefully, identifying the unknowns. b) Assign variables to the unknowns and write down the given information. c) Formulate the quadratic equation based on the problem. d) Solve the equation using appropriate methods. e) Check the solutions in the original problem to ensure they are valid.