Problem 29 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible. $$x^{2}-2 x=1$$

Step-by-Step Solution

Answer

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

Step 1: Rearrangement

First, rearrange the equation so that it equals zero: \(x^{2}-2 x-1=0\). This puts the equation into the standard quadratic form.

Step 2: Apply the Quadratic Formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Apply this formula to the equation. Here, \(a = 1\), \(b = -2\), and \(c = -1\). Substituting these values into the formula yields \[x = \frac{2 \pm \sqrt{(-2)^2 - 4*1*(-1)}}{2*1} = \frac{2 \pm \sqrt{4+4}}{2} = \frac{2 \pm \sqrt{8}}{2}\].

Step 3: Simplify the solution

The square root of 8 can be simplified to \(2\sqrt{2}\) by factoring 8 into \(4* 2\). Simplify the solutions to obtain \(x = 1 \pm \sqrt{2}\).

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations. A quadratic equation is one that can be written in the form:

• \[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
To solve for \( x \), we use the quadratic formula:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula gives us the two possible solutions for \( x \). The term under the square root, \( b^2 - 4ac \), is called the discriminant. It tells us about the nature of the roots:

• If it's positive, there are two real and distinct solutions.
• If it's zero, there is exactly one real solution.
• If it's negative, the solutions are complex numbers.

Understanding and memorizing the quadratic formula can simplify solving quadratic equations.

Simplifying Irrational Solutions

Once you have obtained a solution that involves a square root, it's important to simplify it as much as possible.
Irrational solutions often contain square roots that can be simplified. For example, the square root of 8 can be broken down as follows:

• \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \]

Recognizing such patterns helps in simplifying the expression and making the solutions more presentable.
It's always a good idea to simplify your solutions to their most reduced forms. This often makes it easier to interpret the solutions and shows a better understanding of number properties.

The Process of Solving Equations

Solving equations, especially quadratic ones, involves using logical processes and mathematical tools to find the unknown, \( x \). For quadratic equations, there are various methods such as factoring, completing the square, and using the quadratic formula.
Each method has its pros and cons; however, the quadratic formula is universally applicable to all quadratic equations.
When using the quadratic formula, critical steps include:

• Identifying the coefficients \( a, b, \) and \( c \).
• Substituting them into the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Simplifying the resulting expressions.

Applying systematic procedures helps ensure that no crucial step is missed, leading to accurate and reliable results.

Rearranging Equations for Simplicity

Before applying methods to solve, equations often need to be rearranged into a standard form. This makes the process easier and clearer. When dealing with quadratic equations, the goal is to move all terms to one side of the equality, setting the equation to zero.
For instance, starting with:

• \[ x^2 - 2x = 1 \]
• We can rearrange it to \[ x^2 - 2x - 1 = 0 \].

This sets the stage for using the quadratic formula. Proper rearrangement ensures that the equation is ready for whichever solution method you choose.
Always check that the rearranged equation is accurate and adheres to the requirements of the method you plan to use. Neatly organized equations are much simpler to work with.