Problem 43 - Elementary Algebra

Question

Use the method of extraction of roots to solve \((x-2)^{2}=25\).

Step-by-Step Solution

Answer

Question: Solve the equation \((x-2)^2 = 25\) using the method of extraction of roots. Answer: The solutions to the given equation are \(x=7\) and \(x=-3\).

Step 1: Write down the given equation

We have the equation \((x-2)^2 = 25\).

Step 2: Find the square root of both sides

To find the square root of both sides, we will have: \(\sqrt{(x-2)^2} = \sqrt{25}\). This simplifies to \(|x-2| = 5\).

Step 3: Solve for x using the properties of absolute value

Since \(|x-2| = 5\), we can write two separate equations: \(x-2 = 5\) and \(x-2 = -5\).

Step 4: Solve the first equation for x

Solve the equation \(x-2=5\) for x: \(x=5+2\) \(x=7\)

Step 5: Solve the second equation for x

Solve the equation \(x-2=-5\) for x: \(x=-5+2\) \(x=-3\) Therefore, the solutions to the given equation \((x-2)^2=25\) are \(x=7\) and \(x=-3\).

Key Concepts

Extraction of Roots

The extraction of roots is a fundamental algebraic process used to simplify equations or find unknown variables. This technique involves finding the root of a number or an expression, particularly focusing on square roots or cube roots. Essentially, when you have an equation like \(a^2 = b\), what you're interested in is finding a number \(a\) such that when multiplied by itself gives \(b\).
In the context of this exercise, we dealt with \( (x-2)^2 = 25 \). To solve for \((x-2)\), we extract the square root of both sides:

• Left side gives \(\sqrt{(x-2)^2} = |x-2|\)
• Right side gives \(\sqrt{25} = 5\)

This means that the absolute value of \((x-2)\) is equal to 5, which leads us forward to the consideration of absolute values.

Absolute Value

Absolute value is a concept in algebra that refers to the distance of a number from zero on a number line, regardless of direction. For instance, both -5 and 5 have an absolute value of 5. It's expressed with vertical bars, such as \(|x|\).
In solving \((x-2)^2 = 25\), after taking the square roots, we arrive at \(|x-2| = 5\). This expression tells us that \(x-2\) can either be positive or negative because both \(+5\) and \(-5\) have the same distance from zero.
This leads us to two straightforward cases:

• \(x-2 = 5\) leading to further solving for \(x\)
• \(x-2 = -5\) giving us another solution for \(x\)

By understanding absolute value, you recognize why two equations emerge when a square is involved.

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two. They have the general form \(ax^2 + bx + c = 0\). Our exercise represents a form that can be rewritten and solved by identifying perfect squares.
In the equation \( (x-2)^2 = 25 \):

• The problem is nearly solved with \(x-2\) isolated, indicating the presence of perfect square trinomial.
• Due to the square being isolated, we resort to using the method of extraction to simplify our quadratic equation further.

Once reduced, using the absolute value property \(|x-2|=5\), two linear equations, \(x-2 = 5\) and \(x-2 = -5\), are solved separately. This method represents a simple and effective strategy for handling similar quadratic equations.

Square Roots

Square roots are numbers that produce a specified quantity when multiplied by themselves. For example, the square root of 25 is 5 because \(5 \times 5 = 25\).
In algebra, square roots are often employed to simplify expressions and solve equations. In our exercise, we used square roots to transition from a squared term to an absolute value. The fundamental properties include:

• The square root of a squared variable, \(\sqrt{(x-2)^2}\), results in an absolute value: \(|x-2|\).
• Computing \(\sqrt{25}\), which leads directly to finding those essential solutions.

Understanding square roots is pivotal; they help us progressively dissect quadratic problems by breaking them down into simpler linear elements, capturing both the positive and negative iterations inherent to square numbers.