Problem 41 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-2 x=2 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-2x=2\) are \(x=\(1 + \sqrt{3}\) and \(x=1 - \sqrt{3}\)

Step 1: Rewrite the equation

Start by rewriting the equation \(x^{2}-2x=2\) in the form \(x^{2}-2x-2=0\), by subtracting 2 from both sides of the equation.

Step 2: Complete the square

Next step is to complete the square. To do this, take half the coefficient of x, square it, and add it to both sides of the equation. Here, half of the coefficient of x is -1, and squaring it gives 1. So, the equation becomes \(x^{2}-2x+1=3\), which then simplifies to \( (x-1)^{2}=3 \).

Step 3: Solve the square equation

To solve the square equation, first take the square root of both sides, giving \(x-1=±\sqrt{3}\), and then add 1 to both sides to isolate x: \(x=1±\sqrt{3}\). Therefore the solutions to the equation are \(x=\(1 + \sqrt{3}\) or \(x=1 - \sqrt{3}\) .

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations that helps to transform a quadratic expression into a perfect square trinomial, which is a whole expression squared. This process simplifies solving for the variable. To complete the square, follow these steps smoothly for clarity.

• Identify the coefficient of the linear term. For instance, in the expression \(x^2 - 2x\), the coefficient of \(x\) is \(-2\).
• Calculate half of this coefficient and then square it. In our case, half of \(-2\) is \(-1\), and squaring \(-1\) results in \(1\).
• Add and subtract this square (\(1\)) to the equation to form a perfect square trinomial. It's crucial to add this same value to both sides to keep the equation balanced.

Once you complete the square as shown, it is easier to factor the expression. The equation \(x^2 - 2x + 1 = 3\) simplifies beautifully into \((x-1)^2 = 3\). This indicates that the original trinomials can now be expressed as a square of a binomial, making it easier to solve.

Solving Equations

Solving equations involves finding the values of the variables that satisfy the equation. In our case, after completing the square, our equation \((x-1)^2 = 3\) must be addressed further. It becomes a simple case since the quadratic part is now a perfect square.

To solve this equation efficiently:

• Recognize that \((x-1)^2 = 3\) means you need to "undo" the squaring to solve for \(x\).
• This can be done by applying the method of taking square roots. Bear in mind this approach will yield two results, one for the positive root and another for the negative root based on the square root property.
• Adjusting for the completed square, rearrange terms correctly. This turns into straightforward arithmetic and yields the possible values for \(x\).

In this particular scenario, solving worked effectively through the transformation from quadratic to linear forms, followed by using arithmetic to isolate and solve for the variable.

Square Root Method

The square root method is particularly handy when dealing with equations where one side can be expressed as a perfect square. It enables you to simplify an equation post-completion of the square, or even directly when a term on one side is square of a linear expression.

Using the square root principle in the solved expression \((x-1)^2 = 3\):

• You need to solve \(x-1 = \pm \sqrt{3}\). Notice the plus-minus symbol (±), indicating two potential solutions for the equation due to the nature of squaring losing sign information.
• Proceed by isolating \(x\). Add 1 to both potential forms: First, \(x = 1 + \sqrt{3}\) and second \(x = 1 - \sqrt{3}\).

This traditional approach utilizes the elegance of solving perfect squares using their symmetric properties about the square roots, simplifying otherwise complex algebraic manipulations. Remember, squares can lead to two possible solutions - positive and negative roots, an essential consideration for comprehensive solutions.