Problem 46 - Elementary Algebra

Question

For the following problems, find the products. $$ (y-7.2)^{2} $$

Step-by-Step Solution

Answer

Question: Expand the expression (y - 7.2)^2. Answer: (y - 7.2)^2 = y^2 - 14.4y + 51.84.

Step 1: Identify the formula for binomial squares

For any binomial of the form (a - b)^2, the expanded form can be found using the formula: (a - b)^2 = a^2 - 2ab + b^2

Step 2: Apply the formula to the given expression

In our case, a = y and b = 7.2. Applying the above formula, we get: (y - 7.2)^2 = y^2 - 2(y)(7.2) + (7.2)^2

Step 3: Calculate the coefficients

Now we need to calculate the coefficients of each term: -2(y)(7.2) = -14.4y (7.2)^2 = 51.84

Step 4: Write the final expanded form

The final expanded form of the given expression is: (y - 7.2)^2 = y^2 - 14.4y + 51.84

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical relationships or quantities. In these expressions, variables such as \(y\) act as placeholders that can take on various numerical values. By using algebraic expressions, we can describe and solve problems across many areas of mathematics and real life. Think of them like a language that helps you communicate complex ideas simply and efficiently.

• Exponents denote repeated multiplication. For example, \(y^2\) means \(y\) multiplied by itself.
• Operations include addition (+), subtraction (-), multiplication (×), and division (÷).

Understanding algebraic expressions is key to working with more complex mathematical concepts like equations and functions.

Polynomial Expansion

Polynomial expansion is a process of expressing a power of a binomial as a sum of terms. The binomial theorem provides specific techniques, like binomial squares, to expand expressions such as \((a - b)^2\). When expanding \((a - b)^2\), we apply the formula:
\((a - b)^2 = a^2 - 2ab + b^2\), converting the expression into a simple algebraic form.

• Identify the terms: In \((y - 7.2)^2\), \(a\) is \(y\) and \(b\) is 7.2.
• Calculate each component: Find \(a^2\), \(-2ab\), and \(b^2\).
• Sum these terms to get the expanded expression: \(y^2 - 14.4y + 51.84\).

This expanding allows us to transform a compact expression into a form that easily reveals its structure and properties.

Quadratic Equations

Quadratic equations are polynomials of the second degree, typically written in the form \(ax^2 + bx + c = 0\). Each component of a quadratic equation, the \(x^2\), \(x\), and constant term, plays a crucial role in the equation's properties and solutions.
Quadratic equations can represent various real-world phenomena, such as projectile motion and area calculations.

• Solving methods include factoring, completing the square, and using the quadratic formula.
• The solutions to the equation are referred to as roots and can be real or complex numbers.

Understanding these equations is essential in algebra and advanced mathematics because they frequently arise in practical and theoretical applications.