Problem 17 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ b^{2}=1 $$

Step-by-Step Solution

Answer

Answer: The possible values of \(b\) are \(1\) and \(-1\).

Step 1: Identify the equation type

First, let's identify the type of quadratic equation we are given. The equation \(b^2 = 1\) is a special case of a quadratic equation known as a perfect square trinomial.

Step 2: Rewrite the equation as a perfect square trinomial

We can rewrite the given equation in this form: $$ (b-1)(b+1) = 0 $$

Step 3: Solve for b using the Zero Product Property

Using the Zero Product Property, either the first factor or the second factor must be equal to zero. So, we have two possibilities to solve for \(b\): 1. \(b - 1 = 0\) 2. \(b + 1 = 0\)

Step 4: Find the roots

Now, we find the possible values of \(b\) by solving the equations from Step 3. 1. \(b - 1 = 0 \;\Rightarrow\; b = 1\) 2. \(b + 1 = 0 \;\Rightarrow\; b = -1\)

Step 5: Write the solution

As we found two distinct values of \(b\) that satisfy the given equation, the solution to the quadratic equation \(b^2 = 1\) is: $$b = 1 \;\; \text{or} \;\; b = -1$$

Key Concepts

Extraction of Roots

Extraction of roots is a method used to solve quadratic equations when variables are squared. The process involves taking the square root on both sides of the equation to isolate the variable. This can be applied directly when the equation is of the form \(x^2 = c\). Simply take the square root on both sides:

• \(x = \sqrt{c}\)
• \(x = -\sqrt{c}\)

However, make sure \(c\), the number or expression being squared, is non-negative since we cannot take the square root of a negative number in the real number system. In the case of the original exercise \(b^2 = 1\), by taking the square root of both sides, we find that \(b = 1\) or \(b = -1\). This directly gives us the solution set of the equation. It's essential to understand that the extraction of roots method is particularly efficient when dealing with simple quadratic forms like \(x^2 = k\). Moreover, this method is instrumental in finding real roots, assuming they exist.

Zero Product Property

The Zero Product Property is a crucial concept for solving quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is symbolically represented as:

• If \(a \times b = 0\), then \(a = 0\) or \(b = 0\).

In solving the equation \((b-1)(b+1) = 0\), we apply this property to extract the roots. We assume either factor equals zero since their product is zero:

• \(b-1 = 0\) leads to \(b = 1\)
• \(b+1 = 0\) leads to \(b = -1\)

The Zero Product Property simplifies the process of finding potential solutions by reducing a quadratic equation into simpler linear equations. It's a method often used after rewriting the quadratic equation as a product of binomials, highlighting its importance in factoring and solving polynomials efficiently.

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic expression that can be written as the square of a binomial. It takes the form \(a^2 + 2ab + b^2 = (a+b)^2\) or \(a^2 - 2ab + b^2 = (a-b)^2\). Recognizing this pattern helps in solving quadratic equations more efficiently. In the exercise, the equation \(b^2 = 1\) was not initially in a standard trinomial form, but it can be derived from a factored expression \((b-1)(b+1) = 0\). Here, the binomials are already products recognizing the difference of squares: \(b^2 - 1^2\). This method shows that the expression can be reformulated to a form that's easily factorable. Understanding perfect square trinomials aids in quickly identifying and solving equations by restructuring them into recognized forms. This not only speeds up the process but also reinforces comprehension of factoring techniques and quadratic structures.