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Chapter 1: Problem 33

Solve each equation by the Square Root Method. $$ (x-1)^{2}=4 $$

Short Answer

Expert verified
x = 3 or x = -1

Step by step solution

01

- Isolate the squared term

The given equation is \( (x-1)^2 = 4 \). In this step, the squared term \( (x-1)^2 \) is already isolated on one side of the equation.
02

- Apply the square root to both sides

Take the square root of both sides of the equation. \[ (x-1)^2 = 4 \] \[ \rightarrow \sqrt{(x-1)^2} = \sqrt{4} \] \[ \rightarrow x-1 = \pm2 \]
03

- Solve for x

The equation \((x-1) = \pm2\) leads to two separate linear equations: \((x-1) = 2\) and \((x-1) = -2\). \ To solve each: \For \( (x-1) = 2 \): \[ \rightarrow x = 2 + 1 \rightarrow x = 3 \] \For \((x-1) = -2\): \[ \rightarrow x = -2 + 1 \rightarrow x = -1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Quadratic Equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). These equations can be solved using various methods, one of which is the Square Root Method. The general idea is to simplify the equation into a form where we can take the square root of both sides to solve for the variable. Quadratic equations often have two solutions because of the \(卤\) sign that appears when taking the square root.
If you understand how to isolate the squared term and apply the square root effectively, solving these equations becomes much simpler. Let's look into the core steps now.
Isolating the Squared Term
Isolating the squared term is a crucial step in solving a quadratic equation using the Square Root Method. The squared term is where the variable appears squared, such as \((x-1)^2\). In the given problem, we already have the squared term isolated:
  • The equation starts as \((x-1)^2 = 4\).
  • Here, the squared term \((x-1)^2\) is alone on one side of the equation.

Sometimes, you may need to manipulate the equation to isolate the squared term. This might involve moving other terms to the opposite side of the equation or dividing by a coefficient in front of the squared term. Once the squared term is isolated, the next step is applying the square root to both sides.
Applying Square Roots
After isolating the squared term, you can apply the square root to both sides of the equation. This step helps in removing the exponent and solving for the variable. Let's demonstrate this with our example:
  • Start with \((x-1)^2 = 4\).
  • Apply the square root to both sides: \( \sqrt{(x-1)^2} = \sqrt{4} \).
  • This simplifies to: \(x-1 = 卤2\).

This step introduces the \(卤\) symbol because each number has both a positive and negative square root. So, \( \sqrt{4} \) can be \(2\) or \(-2\). Now, we can solve for the variable simply:
  • For \(x-1 = 2\), adding 1 to both sides gives \(x = 3\).
  • For \(x-1 = -2\), adding 1 to both sides gives \(x = -1\).

Thus, the solutions are \(x = 3\) and \(x = -1\). This perfectly illustrates how isolating the squared term and applying square roots effectively helps in solving quadratic equations.

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