Problem 40 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ u^{2}+u=-4 $$

Step-by-Step Solution

Answer

Answer: The values of 'r' that satisfy the equation are 1 and -1.

Step 1: Identify the base of the equation

In this equation, the base 'r' is raised to the power of 2. The equation is: $$ r^{2} = 1 $$

Step 2: Extract the roots by taking the square root of both sides

To extract the roots, take the square root of both sides of the equation: $$ \sqrt{r^{2}} = \pm\sqrt{1} $$

Step 3: Solve for the variable 'r'

Now, simplify the equation to find the values of 'r': $$ r = \pm 1 $$ The values of 'r' that satisfy the given equation are 1 and -1.

Key Concepts

Extraction of Roots

When solving quadratic equations like \( r^2 = 1 \), one key technique is the extraction of roots. Extraction of roots involves reversing the process of raising a base to a power, specifically by finding the original value of the variable. In our equation, \( r \) is raised to the power of two. To extract the root, we apply the inverse operation of squaring, which is taking the square root.

To correctly extract roots in such equations, always remember these steps:

• Determine the operation applied to the variable (e.g., squaring in this case).
• Apply the opposite operation (e.g., square root) to isolate the variable.
• Consider both the positive and negative roots, because squaring involves multiplying a number by itself, both a positive and a negative number squared yield the same positive result.

By doing so for \( r^2 = 1 \), we find \( r = \pm 1 \). This means both \( r = 1 \) and \( r = -1 \) are solutions. Each value is an extracted root, offering us the solutions to the equation.

Square Roots

Square roots are fundamental to solving quadratic equations where a variable is squared. The square root of a number is a value that, when multiplied by itself, gives the original number. In our problem, we are solving for \( r \) in \( r^2 = 1 \), by finding its square root.

To extract the square roots of an equation like \( r^2 = 1 \), follow these guidelines:

• Compute the square root of the square (i.e., \( \sqrt{r^2} \)). The square root and squaring are inverse operations, thereby simplifying \( r^2 \) to \( r \).
• Apply the square root operation to the entire equation to keep it balanced (i.e., \( \sqrt{r^2} = \pm\sqrt{1} \)), ensuring that you consider the property that \( \sqrt{x^2} = |x| \), resulting in \( r = \pm \sqrt{1} \).

Thus, both \( +1 \) and \( -1 \) are square roots of 1. Always remember, the extraction of square roots requires considering both the plus and minus outcomes.

Algebraic Solutions

The use of algebraic solutions is essential in finding the values of variables in equations. An algebraic solution is simply a method used in math to solve an equation using algebraic manipulations. In our context of \( r^2 = 1 \), we apply algebraic strategies including taking the square root.

To effectively solve algebraically:

• First, identify the form of the equation and the power to which the variable is raised.
• Utilize appropriate operations that reverse that power (e.g., square rooting a square) to isolate the variable.
• Always simplify your results correctly: for \( r^2 = 1 \), simplifying \( \sqrt{r^2} = \pm \sqrt{1} \) gives \( r = \pm 1 \).

The goal with algebraic solutions is to manipulate an equation into a simpler form that reveals the value(s) of unknowns, just like finding \( r = 1 \) and \( r = -1 \) from our equation. This technique is powerful for solving a wide variety of equations in algebra.