Problem 48 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ (x-1)^{2}=4 $$

Step-by-Step Solution

Answer

Answer: The values of x that satisfy the quadratic equation (x-1)^2 = 4 are x = 3 and x = -1.

Step 1: Write down the equation

The given equation is: $$ (x-1)^{2}=4 $$

Step 2: Apply the square root property

To extract the roots of the equation, we will use the square root property, which states that if \(A^2 = B\), then \(A = \pm \sqrt{B}\). Applying this property to our equation, we get: $$ x-1=\pm\sqrt{4} $$

Step 3: Evaluate square root

The square root of 4 is 2, so we can substitute this value into the equation: $$ x-1=\pm 2 $$

Step 4: Solve for x

Add 1 to both sides of the equation to isolate x: $$ x=1\pm 2 $$

Step 5: Find both solutions of x

We have two cases for the values of x: Case 1: When the plus sign is used. $$ x=1+2 $$ $$ x=3 $$ Case 2: When the minus sign is used. $$ x=1-2 $$ $$ x=-1 $$ So, the solutions for the quadratic equation \((x-1)^2 = 4\) are \(x=3\) and \(x=-1\).

Key Concepts

Square Root Property

The square root property is a useful tool for solving certain types of quadratic equations. It allows us to "unwrap" a squared expression by taking the square root of both sides of the equation. This property states that if you have an equation in the form of \( A^2 = B \), you can solve it by expressing \( A \) as \( \pm\sqrt{B} \). This means that the solution will include both the positive and negative square roots.

For example, in the equation \( (x-1)^2 = 4 \), we can apply this property to open up the squared term. By doing so, we convert \( (x-1)^2 \) into \( x - 1 = \pm \sqrt{4} \). This step effectively breaks down the complexity of a squared term, letting us work with simpler linear expressions moving forward in our solution.

Roots of Equations

Roots of equations are the solutions to the equation when set to zero. For quadratic equations, which are polynomial equations of the form \( ax^2 + bx + c = 0 \), these roots can be found using various methods like factoring, completing the square, or utilizing the quadratic formula.

In the context of our exercise, we find the roots by solving the linear equations derived from \( (x-1) = \pm 2 \).

• For \( x - 1 = 2 \), solving gives \( x = 3 \).
• For \( x - 1 = -2 \), solving gives \( x = -1 \).

By solving these equations, we determine the two roots of the original quadratic equation, \((x-1)^2 = 4\). In this scenario, they are \( x = 3 \) and \( x = -1 \). Both values satisfy the original equation when substituted back.

Isolating Variables

Isolating variables is a core strategy in algebra that aims at solving for a specific variable. This involves manipulating an equation to keep the variable of interest on one side, which makes finding its value more straightforward. Typically, this involves addition, subtraction, multiplication, or division to remove coefficients or other terms surrounding the variable.

For instance, once we applied the square root property in the equation \( x - 1 = \pm 2 \), the next step was isolating \( x \). We achieve this by adding 1 to both sides of our expressions:

• \( x - 1 = 2 \rightarrow x = 2 + 1 = 3 \)
• \( x - 1 = -2 \rightarrow x = -2 + 1 = -1 \)

This step is necessary to clearly determine the values of \( x \) that satisfy the equation, leaving us with the solutions \( x = 3 \) and \( x = -1 \). By understanding how to isolate variables, students empower themselves with a critical tool for algebraic problem-solving.