Problem 60 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ x^{2}-2 x-1=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are \(x_1 = 1 + \sqrt{2}\) and \(x_2 = 1 - \sqrt{2}\).

Step 1: Identify a, b, and c

In this quadratic equation, we can see that \(a = 1\), \(b = -2\), and \(c = -1\).

Step 2: Write down the values in the quadratic formula

The quadratic formula is: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ We can insert the values of \(a\), \(b\), and \(c\) to get: $$ x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)} $$

Step 3: Simplify the expression inside the square root

Now, we will simplify the expression inside the square root: $$ x = \frac{2 \pm \sqrt{4 + 4}}{2} $$

Step 4: Calculate the square root and simplify

Let's calculate the square root and simplify the expression: $$ x = \frac{2 \pm \sqrt{8}}{2} $$ You can also write \(\sqrt{8}\) as \(2\sqrt{2}\) since \(8 = 2^2 \cdot 2\): $$ x = \frac{2 \pm 2\sqrt{2}}{2} $$

Step 5: Solve for x

Now, we will divide each term by 2 and solve for \(x\): $$ x_1 = \frac{2 + 2\sqrt{2}}{2} = 1 + \sqrt{2} $$ $$ x_2 = \frac{2 - 2\sqrt{2}}{2} = 1 - \sqrt{2} $$ So, the solutions to the given quadratic equation are \(x_1 = 1 + \sqrt{2}\) and \(x_2 = 1 - \sqrt{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra for solving equations of the form
\[ax^2 + bx + c = 0\],
where \(a\), \(b\), and \(c\) are known values, and \(x\) represents the unknown variable. It is derived from the process of completing the square and can solve any quadratic equation, even when factoring is challenging or impossible. The formula is given by
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
Using this formula involves identifying the coefficients \(a\), \(b\), and \(c\) from the equation and substituting them into the formula. The \(\pm\) symbol indicates that there will usually be two solutions to a quadratic equation, which represents the points where the parabola (the graph of the equation) crosses the x-axis. The process of using the quadratic formula can quickly find these solutions with precision.

Square Root Simplification

Square root simplification is an essential step when solving quadratic equations, especially when working with the quadratic formula. Simplifying the square root involves finding the prime factorization of the number inside the root and identifying pairs of prime factors.
For example, the square root of 8 can be simplified because 8 is a product of prime factors, \(8 = 2^3\). We can take pairs of the same number out of the square root as a single number, so \(\sqrt{8} = \sqrt{2^2 \cdot 2} = 2\sqrt{2}\). Simplifying square roots helps to present the solution in its simplest form and can further facilitate operations such as addition, subtraction, multiplication, or division involving square roots.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). They do not have an equality sign, unlike equations. When dealing with quadratic equations, the expressions often include the variable \(x\) raised to the second power, indicated as \(x^2\), defining them as quadratic.
In the context of solving quadratic equations using the quadratic formula, the expression under the square root sign (\(b^2 - 4ac\)) is crucial as it determines the nature of the roots. If the expression is positive, the quadratic equation has two real and distinct solutions. If it's zero, there is exactly one real solution. And if the expression is negative, the equation has two complex solutions. Understanding algebraic expressions and how to manipulate them is fundamental to working with quadratic equations.

Quadratic Equation Solutions

The solutions of a quadratic equation are the values of the variable that make the equation true. These solutions can be real or complex and are the points where the parabola (the curve representing the quadratic equation) intersects the x-axis.
In the exercise \(x^2 - 2x - 1 = 0\), using the quadratic formula provides two solutions: \(x_1 = 1 + \sqrt{2}\) and \(x_2 = 1 - \sqrt{2}\). These solutions are derived after simplifying the expression under the square root and dividing each term by the coefficient of \(x^2\), which in this case, is 1. The ability to find these solutions and understand their meaning is a central part of algebra and is vital for many fields of mathematics, science, and engineering.