Problem 19 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ r^{2}=25 $$

Step-by-Step Solution

Answer

Answer: The roots of the quadratic equation are r = +5 and r = -5.

Step 1: Identify the equation

We are given the quadratic equation: $$ r^2 = 25 $$

Step 2: Take the square root of both sides

In order to solve for the variable r, we will take the square root of both sides of the equation: $$ \sqrt{r^2} = \sqrt{25} $$

Step 3: Simplify and find the roots

Simplifying the equation, we get: $$ r = \pm \sqrt{25} $$ So, the roots of the given equation are: $$ r = +5 \text{ and } r = -5 $$

Key Concepts

extraction of roots

The extraction of roots is a mathematical technique used to solve equations that involve powers or exponents. In our exercise, we are dealing with a square, which is an expression raised to the power of 2. To extract the roots means finding the original number that was squared to reach our result. When an equation takes the form \( r^2 = 25 \), extracting the root involves determining the values of \( r \) such that when squared, they equal 25.
To extract roots, we:

• Identify the equation and ensure it is in a form that allows taking square roots, such as \( r^2 = 25 \).
• Apply the square root operation to both sides. This means calculating \( \sqrt{r^2} = \sqrt{25} \), which simplifies to \( r = \pm 5 \).

The concept of extracting roots in this way is vital for solving quadratic equations efficiently, particularly when they are set equal to a perfect square.

quadratic equation

A quadratic equation is an equation that includes a variable raised to the power of two as its highest exponent. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. However, the equation we solved, \( r^{2} = 25 \), is a simpler form known as a pure quadratic equation because the term with \( r \) (the linear term) and the constant are isolated on different sides br> Quadratic equations can have:

• Two solutions: which are referred to as roots, such as \( r = +5 \) and \( r = -5 \).
• One solution: when the discriminant of the quadratic equation is zero.
• No real solution: when the discriminant is negative and solutions are complex numbers.

By simplifying \( r^2 = 25 \) through the method of extraction of roots or other methods like factoring and completing the square, we arrive at these solutions.

square root method

The square root method is a straightforward technique used particularly for solving quadratic equations without a linear term. This method becomes useful when the equation is simplified to the form \( x^2 = k \), where \( k \) is a constant. It directly utilizes the property of square roots by reversing the squaring process.
Here's how it's applied:

• Ensure the quadratic equation is in a simple form, such as \( r^2 = k \).
• Take the square root of both sides, making sure to consider both the positive and negative roots to solve for \( r \).
• In our exercise, we identified the roots \( r = \pm 5 \) by calculating \( \sqrt{25} \).

Using the square root method is not only limited to purely quadratic equations with perfect squares but also adapts when dealing with radical expressions. It's a valuable strategy when factoring or other common techniques are less manageable.