Problem 41 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by completing the square. $$x^{2}-2 x-2=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 1 + \sqrt{3} \) and \( x = 1 - \sqrt{3} \).

Step 1: Move the constant term

Start by moving the constant term to the other side of the equation. This gives you:\[ x^2 - 2x = 2 \]

Step 2: Complete the square

To complete the square, add and subtract the square of half the coefficient of the x-term inside the equation. The coefficient of the x-term is -2. Half of -2 is -1, and its square is 1. Add and subtract 1 inside the equation:\[ x^2 - 2x + 1 - 1 = 2 \]Which simplifies to:\[ (x - 1)^2 - 1 = 2 \]

Step 3: Move the extra constant term

Move the subtracted 1 to the right side to isolate the square term:\[ (x - 1)^2 = 3 \]

Step 4: Solve for x

Take the square root of both sides to solve for x:\[ x - 1 = \pm\sqrt{3} \]Then solve for x by isolating it:\[ x = 1 \pm\sqrt{3} \]

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two. Basically, they take the form:
\[ ax^2 + bx + c = 0 \]
Where a, b, and c are constants, with a ≠ 0. When dealing with quadratic equations, we often aim to find the value(s) of x that satisfy the equation.

One common approach to solve these equations is by factoring. However, not all quadratics are easy to factor. That's when methods like completing the square come in handy. Completing the square transforms the quadratic equation into a perfect square trinomial, making it easier to apply further algebraic techniques.

Algebraic Methods

Algebraic methods are strategies used to manipulate equations and expressions so that they reveal the information we are looking for. With quadratic equations, we commonly use methods such as:

• Factoring
• Using the Quadratic Formula
• Completing the Square

For our specific exercise, we focus on completing the square. Let's outline the steps in detail:

• Step 1: Move the constant term - We start by isolating the quadratic and linear terms on one side of the equation.
  
• Step 2: Complete the square - Add and subtract the square of half the coefficient of the x-term. This creates a perfect square trinomial on one side of the equation.
• Step 3: Move the extra constant term - Isolate the square term by moving the subtracted amount to the other side of the equation.


• Step 4: Solve for x - Finally, take the square root of both sides and solve for x.

By breaking down the problem into these approachable steps, we can methodically solve the equation.

Solving Equations

Solving equations entails finding the value(s) that satisfy the given mathematical statement. For quadratic equations, completing the square is especially useful when the equation does not factor easily. Let's recap the solution process for the provided exercise:

• We begin with \( x^2 - 2x - 2 = 0 \) and isolate the quadratic and linear terms: \( x^2 - 2x = 2 \).


• Next, we complete the square. We take half of the coefficient of x (which is -2), square it to get 1, and then add and subtract this inside the equation: \( x^2 - 2x + 1 - 1 = 2 \), simplifying to: \( (x - 1)^2 - 1 = 2 \).


• We then move the extra constant term to the right side to isolate the perfect square trinomial: \( (x - 1)^2 = 3 \).


• Taking the square root of both sides, we solve for x: \( x - 1 = \pm \sqrt{3} \), leading to solutions \( x = 1 + \sqrt{3} \) and \( x = 1 - \sqrt{3} \).

Each step builds on the previous one, emphasizing the importance of a structured approach to solve complex problems.