Problem 35 - Basic Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Structure

The given equation \((x+a)^2-b^2=0\) is a difference of squares. Recall that a difference of squares has the form \(A^2-B^2\) and is factored as \((A-B)(A+B)\). In this problem, \(A = (x+a)\) and \(B = b\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula, factor \((x+a)^2-b^2\) as \(((x+a)-b)((x+a)+b)\). This gives us the equation: \(((x+a)-b)((x+a)+b)=0\).

Step 3: Set Each Factor to Zero

To find the solutions for \(x\), set each factor equal to zero: 1. \((x+a)-b = 0\)2. \((x+a)+b = 0\)

Step 4: Solve Each Equation

For the first equation \((x+a) - b = 0\), solve for \(x\): \[x + a - b = 0\]Subtract \(a\) from both sides:\[x - b = -a\]Add \(b\) to both sides:\[x = b - a\]For the second equation \((x+a) + b = 0\), solve for \(x\):\[x + a + b = 0\]Subtract \(a\) from both sides:\[x + b = -a\]Subtract \(b\) from both sides:\[x = -a - b\]

Step 5: State the Solutions

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

Key Concepts

Factoring Techniques

Factoring techniques are essential in solving quadratic equations and can often lead to simpler solutions. In our exercise, the main technique used is recognizing and applying the "difference of squares" formula. When you encounter a quadratic expression that looks like a difference of squares, such as \((x+a)^2 - b^2\), it can be factored into two binomials:

• \((x+a - b) \)
• \((x+a + b) \)

This technique relies on a deeper understanding of algebraic identities, where the difference of squares allows for easier manipulation and resolution of the equation into linear factors.
What's great about factoring techniques is that they transform complex polynomials into manageable parts, making it easier to find roots or solutions.

Difference of Squares

The difference of squares is a crucial algebraic identity used in factoring. This identity states that any expression of the form \(A^2 - B^2\) can be rewritten as \((A - B)(A + B)\).
In our example, \((x+a)^2 - b^2 = 0\), we identified it as a difference of squares by comparing it to the standard formula. Here, \(A = (x+a)\) and \(B = b\).Recognizing this pattern allows us to break the equation down:

• This results in two simpler linear expressions, \((x+a) - b\) and \((x+a) + b\).

By following this method, solving complex quadratic equations becomes a straightforward task, utilizing the elegance and symmetry inherent in algebraic expressions.

Solving Equations

Solving equations is a fundamental component of algebra. After factoring a given quadratic equation, the next step is to find the solutions by setting each factor equal to zero.
Our factored equation \(((x+a) - b)((x+a) + b) = 0\) is simplified to:

• \((x+a) - b = 0\)
• \((x+a) + b = 0\)

Each of these linear equations can be solved separately to find the values of \(x\) that satisfy the original quadratic equation.To solve \((x+a) - b = 0\): First, isolate \(x\) by removing terms on one side, typically starting with subtraction or addition. In our case:\[x - b = -a\]Then, add \(b\) to both sides to solve for \(x\):\[x = b - a\]Similarly, solve \((x+a) + b = 0\) by adjusting terms:\[x = -a - b\]These solutions are found because setting the factors to zero simplifies the equation and reveals the possible values for \(x\).

Mathematical Problem Solving

Mathematical problem solving involves a series of strategic steps. It often starts with identifying the type of problem you are facing and applying suitable methods to solve it. In quadratic equations, like this exercise, identifying the equation's structure (for instance, as a difference of squares) is key.
The process involves:

• Recognizing patterns, such as factoring techniques or identities.
• Applying the correct algebraic formulas and methods.
• Breaking down the problem into smaller, manageable parts.
• Finding solutions by setting parts equal to zero and solving these simple equations.

Each step requires a basic understanding of algebra and attention to detail. Catching mistakes or missteps early in the problem solving process helps achieve accurate results. Engaging in practice with different problems enhances your skills, making you proficient in tackling various algebraic equations and more complex mathematical challenges.