Problem 34 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ 5 b^{2}-5=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the given quadratic equation are \(b = 1\) and \(b = -1\).

Step 1: Rewrite the equation

First, rewrite the given quadratic equation in the standard form: $$ 5b^2 - 5 = 0 $$

Step 2: Isolate the b^2 term

Divide both sides of the equation by 5 in order to isolate the \(b^2\) term: $$ b^2 - 1 = 0 $$

Step 3: Write as a difference of squares

Express the equation as the difference of two squares. The equation \(b^2 - 1 = 0\) can be written as: $$ (b - 1)(b + 1) = 0 $$

Step 4: Solve using extraction of roots

Now, apply the extraction of roots method to find the two possible solutions for b: Case 1: \((b - 1) = 0\) $$ b = 1 $$ Case 2: \((b + 1) = 0\) $$ b = -1 $$ Thus, the solutions for the given quadratic equation are \(b = 1\) and \(b = -1\).

Key Concepts

Extraction of Roots

Extraction of roots is a straightforward method used to solve quadratic equations when they are already set up as a perfect square. In this process, the main goal is to isolate the variable squared on one side of the equation and then take the square root to solve for the variable. Let's break it down using our given equation:

• Start by ensuring the quadratic is in the form that easily separates the squared term, like \(b^2 = k\).
• In the exercise, after dividing by 5, we get \(b^2 = 1\).
• Taking the square root of both sides, we need to remember that roots can be both positive and negative.
• This yields \(b = \pm 1\).

Therefore, the technique hinges on breaking down the square to find the variables, making it one of the quicker methods for quadratics that simplify easily to a square form.

Solving Quadratics

Solving quadratic equations is a fundamental skill in algebra. Quadratics are equations with the general form \(ax^2 + bx + c = 0\). There are several methods, but the key idea is to determine the values of the variable that make the equation true. Here's a brief overview:

• The methods include factoring, using the quadratic formula, completing the square, and graphing.
• In the problem earlier, factoring was used implicitly after expressing the quadratic as \((b - 1)(b + 1) = 0\).
• Once in factorable form, the zero-product property states if \(ab = 0\), then \(a = 0\) or \(b = 0\).
• Apply this property to arrive at each possible solution, as shown with \(b = 1\) or \(b = -1\).

Each step in solving quadratics should aim to simplify the equation to reveal the variable’s possible values. The technique used depends on the given equation's complexity and form.

Difference of Squares

The difference of squares is a special algebraic technique used to factor quadratic expressions. This method is applicable when you have two perfect squares separated by a subtraction sign, and it's very useful for solving quadratics. Let's look deeper into this concept:

• A difference of squares takes the form \(a^2 - b^2\) and can be factored into \((a - b)(a + b)\).
• In the exercise, the equation \(b^2 - 1 = 0\) can be viewed as \((b^2) - (1^2)\), fitting the difference of squares form.
• This transformation allows quick factorization, leading to the factorized form \((b - 1)(b + 1) = 0\).
• It's an efficient way to solve the quadratic as shown in the step-by-step solution.

Understanding how to identify and use the difference of squares can simplify solving equations significantly, especially when the quadratic doesn't fit more complex factoring needs.