Problem 103 - Elementary Algebra

Question

Create an equation of your own that can be solved by extracting the root. Share it, along with the solution, on the discussion board.

Step-by-Step Solution

Answer

Equation: \( x^2 - 9 = 0 \). Solutions: \( x = 3 \) and \( x = -3 \).

Step 1: Define the Equation

Let's start by writing down a quadratic equation of our own: \( x^2 - 9 = 0 \). This equation is set up such that we can solve it by finding the square root.

Step 2: Isolate the Square Term

To solve the equation \( x^2 - 9 = 0 \), first, isolate the term with the square. Add 9 to both sides of the equation: \( x^2 = 9 \).

Step 3: Extract the Square Root

To find \( x \), take the square root of both sides of the equation. This results in \( x = \pm \sqrt{9} \).

Step 4: Simplify the Solution

The square root of 9 is 3. Therefore, \( x = \pm 3 \). This means we have two solutions: \( x = 3 \) and \( x = -3 \).

Key Concepts

extracting roots

In mathematics, extracting roots is a technique used to find the solution of an equation where a variable is raised to a power. In the context of solving quadratic equations, this involves determining the value of a variable when the variable itself is squared.
One simple example is the equation \( x^2 = 9 \). Here, we have a quadratic equation where the variable, \( x \), is raised to the power 2. By extracting the root, we're looking for values of \( x \) that satisfy this equation.

• Understand that the equation representation is very important: the term squared (\( x^2 \)) should be isolated.
• A root of a squared term is simply the reverse operation of squaring and can be calculated here by using the square root method.
• Be aware of the concept of positive and negative roots i.e., the value of \( x \) can be either positive or negative, since both satisfy \( x^2 = 9 \): \( x = 3 \) and \( x = -3 \).

By extracting roots, we reveal the solutions to the equation directly and effectively, as it reduces a potentially complex operation to the simple calculation of a square root.

isolating terms

To solve an equation effectively, it's essential to isolate the term that involves the variable you seek to solve for. Isolating terms is a fundamental algebraic technique that involves rearranging an equation to make the desired variable the subject.
Consider the quadratic equation \( x^2 - 9 = 0 \). Here, we aim to isolate \( x^2 \) so that we can solve for \( x \).

• Start by moving all other terms to the opposite side of the equation. For \( x^2 - 9 = 0 \), add 9 to both sides to isolate \( x^2 \): \( x^2 = 9 \).
• This step is fundamental because it prepares the equation for applying the square root method, which facilitates solving for \( x \).
• Always ensure that your operations maintain the balance of the equation for it to remain true.

Isolating the term allows us to transform a potentially tangled problem into a simplified form that is much more straightforward to solve.

square root method

The square root method is a straightforward algebraic technique used to solve equations of the form \( x^2 = c \), where \( c \) is a constant. This method involves taking the square root of both sides of the equation to find the values of \( x \).
Using the example \( x^2 = 9 \), we apply the square root to both sides to solve for \( x \):

• The calculation becomes \( x = \pm \sqrt{9} \).
• Remember that the square root of a number gives two values: one positive and one negative (e.g., \( \sqrt{9} = 3 \) and \( -3 \)).
• Thus, the solutions are \( x = 3 \) and \( x = -3 \), representing the two possible values that satisfy the original equation.

The understanding of the square root method is crucial because it provides a clear approach to dealing with quadratic equations where the term is isolated. This method simplifies solving by turning the process into basic arithmetic: calculating square roots.