Finding The Product Of (n+7)(n-2) A Mathematical Exploration
In the realm of mathematics, algebraic expressions serve as fundamental building blocks for representing and solving a wide range of problems. These expressions, composed of variables, constants, and mathematical operations, allow us to model real-world scenarios, explore relationships between quantities, and develop powerful problem-solving techniques. One common task in algebra involves finding the product of two or more expressions, which essentially means multiplying them together. This process often leads to simplification and reveals underlying patterns within the expressions.
Expanding the Product: A Step-by-Step Approach
To find the product of the expressions (n + 7)(n - 2), we employ the distributive property, a cornerstone of algebraic manipulation. The distributive property states that multiplying a sum by a number is the same as multiplying each addend individually by the number and then adding the products. In our case, we'll apply this property twice, once for each term in the first expression.
Let's break down the process:
- Distribute the first term (n) of the first expression:
- Multiply n by each term in the second expression: n(n - 2) = n * n* - n * 2 = n^2 - 2n
- Distribute the second term (7) of the first expression:
- Multiply 7 by each term in the second expression: 7(n - 2) = 7 * n - 7 * 2 = 7n - 14
- Combine the results:
- Add the results from steps 1 and 2: (n^2 - 2n) + (7n - 14)
- Simplify by combining like terms:
- Identify terms with the same variable and exponent: n^2, -2n, 7n, and -14
- Combine the 'n' terms: -2n + 7n = 5n
- Write the final expression: n^2 + 5n - 14
Therefore, the product of (n + 7)(n - 2) is n^2 + 5n - 14. This resulting expression is a quadratic polynomial, characterized by the highest power of the variable being 2.
The expression n^2 + 5n - 14 represents a quadratic function, a fundamental concept in mathematics with wide-ranging applications. Quadratic functions are used to model various phenomena, such as the trajectory of a projectile, the shape of a parabola, and the growth of certain populations.
The product we found, n^2 + 5n - 14, can be further analyzed to reveal key characteristics of the corresponding quadratic function. For instance, we can determine the roots (or zeros) of the function by setting the expression equal to zero and solving for n. These roots represent the points where the graph of the function intersects the x-axis.
Finding the Roots: Factoring the Quadratic Expression
To find the roots of n^2 + 5n - 14, we can attempt to factor the expression. Factoring involves breaking down the quadratic into two linear expressions, which, when multiplied, give the original quadratic. In this case, we're looking for two numbers that multiply to -14 and add to 5. These numbers are 7 and -2.
Therefore, we can factor the expression as follows:
- n^2 + 5n - 14 = (n + 7)(n - 2)
Now, to find the roots, we set each factor equal to zero and solve for n:
- n + 7 = 0 => n = -7
- n - 2 = 0 => n = 2
Thus, the roots of the quadratic function represented by n^2 + 5n - 14 are -7 and 2. These roots correspond to the points (-7, 0) and (2, 0) on the graph of the function.
The graph of a quadratic function is a parabola, a U-shaped curve that is symmetrical about a vertical line called the axis of symmetry. The roots of the function, which we found to be -7 and 2, represent the x-intercepts of the parabola. The vertex, the point where the parabola changes direction, is another important feature of the graph.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic expression in the form ax^2 + bx + c. In our case, a = 1 and b = 5, so the x-coordinate of the vertex is x = -5/(2*1) = -2.5.
To find the y-coordinate of the vertex, we substitute x = -2.5 into the expression n^2 + 5n - 14:
- y = (-2.5)^2 + 5(-2.5) - 14 = 6.25 - 12.5 - 14 = -20.25
Therefore, the vertex of the parabola is at the point (-2.5, -20.25). Knowing the roots and the vertex allows us to sketch a reasonably accurate graph of the quadratic function. The parabola opens upwards since the coefficient of the n^2 term is positive.
Quadratic functions have numerous applications in various fields, including physics, engineering, economics, and computer science. Some examples include:
- Projectile Motion: The trajectory of a projectile, such as a ball thrown in the air, can be modeled using a quadratic function. The function can be used to determine the maximum height reached by the projectile and the distance it travels.
- Optimization Problems: Quadratic functions can be used to find the maximum or minimum values of certain quantities. For example, a company might use a quadratic function to determine the optimal price for a product to maximize profit.
- Curve Fitting: Quadratic functions can be used to fit curves to data points. This technique is used in various applications, such as data analysis and modeling.
- Engineering Design: Quadratic functions are used in the design of bridges, arches, and other structures.
Finding the product of algebraic expressions like (n + 7)(n - 2) is a fundamental skill in algebra. By applying the distributive property and simplifying the resulting expression, we can gain insights into the underlying relationships between variables and constants. In this case, we found that the product, n^2 + 5n - 14, represents a quadratic function, which has numerous applications in various fields. Understanding quadratic functions and their properties is crucial for solving a wide range of mathematical and real-world problems. By mastering algebraic manipulation techniques and exploring the significance of the results, we can unlock the power of mathematics to model and understand the world around us.
- Quadratic Expression
- Algebraic Expression
- Distributive Property
- Factoring Quadratics
- Roots of a Quadratic Function
- Parabola
- Vertex of a Parabola
- Applications of Quadratic Functions
- Mathematical Problem Solving
- Algebra Fundamentals
