Problem 100 - Precalculus Essentials

Question

Solve by completing the square: $$ x^{2}-2 x-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 1 + \sqrt{2} \) and \( x = 1 - \sqrt{2} \)

Step 1: Rearrange the Equation

Rearrange the equation to make the \( x^2 \) and \( x \) terms on one side, and the constant on the other side. The equation becomes \( x^2 - 2x = 1 \)

Step 2: Complete the Square

To convert the left-hand side into a perfect square, add the square of half the coefficient of the \( x \) term to both sides of the equation. Half of -2 is -1, and its square is 1. After adding this, the equation becomes \( (x - 1)^2 = 1 + 1 \) which simplifies to \( (x - 1)^2 = 2 \)

Step 3: Solve for x

Take the square root of both sides. Remember that the square root of a number is both its positive and negative square root. Thus, the solutions are \( x - 1 = \sqrt{2} \) and \( x - 1 = -\sqrt{2} \). Solving these separately for \( x \), we get \( x = 1 + \sqrt{2} \) and \( x = 1 - \sqrt{2} \)

Key Concepts

Quadratic Equations

Quadratic equations are polynomials of degree two and can be expressed in the general form \( ax^2 + bx + c = 0 \). In our exercise, the equation given is \( x^2 - 2x - 1 = 0 \), which fits this form with \( a = 1 \), \( b = -2 \), and \( c = -1 \).
When solving quadratic equations, there are various methods available: factoring, using the quadratic formula, graphing, or completing the square. The method of completing the square is a valuable technique because it transforms the equation into a form that is much easier to solve, particularly useful for equations that do not factor easily.
Understanding the components of quadratic equations and the different methods to solve them is essential. This enhances one’s ability to tackle a variety of problems, especially those that appear complex at first glance.

Square Root

The square root is a mathematical operation that finds the original number that was squared to get a given value. For example, the square root of 4 is 2 because \( 2^2 = 4 \). In the context of solving quadratic equations, taking the square root is a crucial step.
In our problem, after converting the equation into a perfect square, we reach the expression \( (x - 1)^2 = 2 \). To solve for \( x \), we take the square root of both sides, leading to two possible values: \( x - 1 = \sqrt{2} \) and \( x - 1 = -\sqrt{2} \).
This step is important because it yields two solutions due to the fact that both a positive and a negative number squared will give the same result. Remember, whenever you take the square root of both sides of an equation, include the \( \pm \) symbol to account for both solutions.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression of the form \( (x + a)^2 = x^2 + 2ax + a^2 \). This concept is pivotal when completing the square. By transforming a quadratic expression into a perfect square trinomial, we simplify the solving process.
In the exercise, by taking half of the coefficient of \( x \) (which was -2, thus -1), and squaring it, we added 1 to both sides of the equation. This transformed \( x^2 - 2x \) into the expression \( (x - 1)^2 \).
Perfect square trinomials allow us to write quadratic equations in a way that makes it straightforward to solve using the square root method. Recognizing and converting expressions into perfect squares is a valuable skill that simplifies complex quadratic problems.