Problem 18 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$1-x^{2}=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -1\).

Step 1: Recognize the Equation Type

The equation given is \(1 - x^2 = 0\). This is a quadratic equation which can be rewritten as \(-x^2 + 1 = 0\) or \(1 - x^2 = 0\). We notice that this equation is a difference of squares.

Step 2: Apply Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). In our equation, we have \(1 - x^2 = (1)^2 - (x)^2\). Therefore, it factors into \((1-x)(1+x) = 0\).

Step 3: Solve Each Factor

Set each factor equal to zero: 1. \((1-x) = 0\)2. \((1+x) = 0\)Solve for \(x\) in each equation.

Step 4: Solve for x in (1-x)=0

For the equation \(1-x = 0\), add \(x\) to both sides to get \(1 = x\). So, one solution is \(x = 1\).

Step 5: Solve for x in (1+x)=0

For the equation \(1+x = 0\), subtract 1 from both sides to get \(x = -1\). Thus, the other solution is \(x = -1\).

Step 6: Final Solution

The solutions to the equation \(1-x^{2}=0\) are \(x = 1\) and \(x = -1\).

Key Concepts

Factoring Techniques

Understanding factoring techniques is crucial for solving quadratic equations efficiently. Factoring is simply the process of breaking down an expression into simpler terms, or factors, that, when multiplied together, give the original expression. The key to successful factoring is recognizing different patterns or using specific techniques.

• Recognize Patterns: A common pattern is the difference of squares, where the expression looks like \(a^2 - b^2\).
• Utilize Common Factors: First, always look for a greatest common factor (GCF) that can be factored out of an equation. This can simplify the equation and make further factoring easier.

Factoring is an essential skill in algebra and requires practice to spot these patterns and determine the best method to apply. In the case of simple quadratic equations, recognizing and applying these patterns can quickly lead to discovering the roots of the equation.

Difference of Squares

The difference of squares is a specific factoring technique used for expressions of the form \(a^2 - b^2\). This pattern is unique because it can be easily factored into a pair of binomials:(a-b)(a+b).This works due to the relationship between squares, where the middle terms cancel each other out:

• The expanded form \((a-b)(a+b)\) results in \(a^2 - b^2\) because the \(-ab\) and \(+ab\) terms cancel.
• This makes the equation easier to solve as it provides two simpler linear equations after factoring.

For the example \(1-x^2=0\), we recognize it as \((1)^2 - (x)^2\). Factoring it as \((1-x)(1+x)=0\), simplifies finding solutions because we now solve two linear equations rather than one quadratic.

Solving Equations Step by Step

Solving quadratic equations using factoring involves a clear, step-by-step process. This ensures no steps are missed and results are accurate. Here are the typical steps:1. Identify the equation type: Determine if the equation can be factored using a recognizable pattern, such as the difference of squares in our example.2. Apply the appropriate factorization technique: Depending on the pattern, rewrite the equation in its factored form. For \(1-x^2=0\), we used the difference of squares.3. Set each factor to zero: Break down the factored equation into simpler parts. For \((1-x)(1+x)=0\), split into two equations: \(1-x=0\) and \(1+x=0\).4. Solve each equation:

• For \(1-x=0\), adding \(x\) to both sides gives \(x=1\).
• For \(1+x=0\), subtracting \(1\) from both sides results in \(x=-1\).

5. Conclude with solutions: The solutions you find are the values that satisfy the original equation. For this example, they are \(x=1\) and \(x=-1\).By following these steps, solving quadratic equations can be systematic and relatively straightforward, especially when common patterns like the difference of squares are recognized early.