Problem 35 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$(x+a)^{2}-b^{2}=0$$

Step-by-Step Solution

Answer

The solutions are \(x = b-a\) and \(x = -a-b\).

Step 1: Recognize the Difference of Squares

The equation \((x+a)^2 - b^2 = 0\) is in the form of a difference of squares, which can be expressed as \(A^2 - B^2 = (A-B)(A+B)\). Here, \(A = x+a\) and \(B = b\).

Step 2: Apply the Difference of Squares Formula

Using the formula \((A-B)(A+B)\), we can write the equation as \((x+a-b)(x+a+b) = 0\).

Step 3: Solve for x

To solve the equation \((x+a-b)(x+a+b) = 0\), we set each factor to zero: 1. \(x+a-b = 0\) which gives \(x = b-a\). 2. \(x+a+b = 0\) which gives \(x = -a-b\).

Step 4: Write the Final Solution

The solutions for the quadratic equation \((x+a)^2 - b^2 = 0\) are \(x = b-a\) and \(x = -a-b\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebra, used for factoring certain quadratic expressions. When we talk about the difference of squares, we're referring to an expression that takes the form \(A^2 - B^2\). This can be rewritten using the formula: \((A-B)(A+B)\). This might look a bit abstract at first, but it becomes clearer with numbers.
For example, consider the expression \(9 - 4\). This is equivalent to \(3^2 - 2^2\), and using the difference of squares formula, we can factor it as \((3-2)(3+2)\) or \((1)(5)\). What's convenient about the difference of squares is that it allows us to break down complex expressions into simpler factors, making them easier to solve.
In the original exercise you were given

1. \( (x+a)^2 - b^2 = 0 \), with \( A = x+a \) and \( B = b \).

By recognizing the structure of the equation as a difference of squares, we can apply the formula to factor and eventually solve for \(x\). This approach simplifies the process of solving quadratic equations by leveraging the unique properties of squares.

Solving Quadratic Equations

When solving quadratic equations, factoring is one of the most straightforward methods to tackle them. A quadratic equation is generally given in the form \(ax^2 + bx + c = 0\). However, the equation from the original exercise, \((x+a)^2 - b^2 = 0\), is a special scenario where factoring can be performed using the difference of squares.
Once we recognize the difference of squares in the equation, we apply the formula to get our factors: \((x+a-b)(x+a+b) = 0\). This is a classic technique because setting each factor to zero allows us to find the values of \(x\) that solve the equation.

• \(x+a-b=0\): Solve this to find \(x = b-a\).
• \(x+a+b=0\): Solve this to get \(x = -a-b\).

This gives us the final solutions for the equation. This technique reduces a complex expression into linear factors, providing an efficient way to determine the solutions to quadratic equations.

Basic Algebra Concepts

Before diving into solving quadratic equations, it's essential to grasp some basic algebra concepts. These fundamental ideas form the building blocks for understanding and solving more complex mathematical problems. In algebra, we often work with expressions that involve variables, constants, and operations such as addition, subtraction, multiplication, and division.
Variables like \(x, a,\) and \(b\) are symbols used to represent unknown numbers or values in equations. Constants are fixed values, such as \(2, 3,\) or any specific number, that do not change. Operations on these components help us manipulate equations to isolate variables and solve for them. Understanding how to rearrange terms and apply algebraic formulas is crucial for solving mathematical expressions effectively.

• Expressions: These are combinations of variables and numbers, like \((x+a)^2\), and can often be simplified or factored.
• Equations: Algebraic equations, such as \((x+a)^2 - b^2 = 0\), are statements that two expressions are equal.

These basic concepts of algebra allow us to build equations, transform them using various methods, and eventually solve them to find the unknown values. When you're comfortable with these foundations, tackling quadratic equations becomes significantly easier.