Problem 48 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{2}=18 $$

Step-by-Step Solution

Answer

The real solutions are \(x = 3\sqrt{2}\) and \(x = -3\sqrt{2}\).

Step 1: Understand the equation

The given equation is quadratic: \(x^2 = 18\). Our task is to find the values of \(x\) that satisfy this equation.

Step 2: Take the square root of both sides

To solve for \(x\), we need to eliminate the square on the left side. We do this by taking the square root of both sides of the equation: \(x = \pm \sqrt{18}\).

Step 3: Simplify the square root

The square root \(\sqrt{18}\) can be simplified further since 18 can be expressed as a product of its prime factors: \(18 = 9 \times 2 = 3^2 \times 2\). This gives us \(\sqrt{18} = \sqrt{3^2 \times 2} = 3\sqrt{2}\). Thus, \(x = \pm 3\sqrt{2}\).

Key Concepts

Real Solutions in Quadratic Equations

In the context of quadratic equations, the term 'real solutions' refers to the values of the variable that satisfy the equation, resulting in real numbers. A quadratic equation can have zero, one, or two real solutions. To determine these, especially when dealing with equations like \(x^2 = 18\), we need to analyze the equation's structure. In our given equation, we are interested in finding the values of \(x\) that, when squared, equal 18, which corresponds to the points where the parabola intersects the x-axis.

• If the equation has two distinct real solutions, then the variable \(x\) can take two values.
• These solutions are found by considering both the positive and negative square roots of the number equated (18 in this instance).

The term "+-" before a square root, such as \(\pm \sqrt{18}\), indicates that there are two solutions: one positive and one negative, because both will satisfy \(x^2=18\).

Understanding and Using the Square Root

The square root is a fundamental concept when dealing with quadratic equations. It is the inverse operation of squaring a number. Taking the square root allows us to solve equations involving squares. When we encounter an equation like \(x^2 = 18\), our goal is to take the square root of both sides to "undo" the squaring operation.

• The notation \(\sqrt{18}\) represents a number which, when multiplied by itself, equals 18.
• The square root of a number always has two values: a positive and a negative, due to the nature of squaring (e.g., \((+3)^2 = 9\) and \((-3)^2 = 9\)).

This concept is vital as it underlines the fact that quadratic equations typically do not just have a single solution, but rather a pair of solutions, unless specified otherwise. Hence, we represent these solutions using the plus-minus sign \(\pm\) in our solution, indicating both possibilities.

Simplifying Radicals

Simplifying radicals involves breaking down a square root into its simplest form. This often requires recognizing the number inside the radical as a product of perfect squares and other factors. ### Breaking Down RadicalsWhen faced with \(\sqrt{18}\), one can simplify it by factoring 18 into its components: 18 equals \(9 \times 2 = 3^2 \times 2\). The \(3^2\) part can be taken out of the square root as 3, since \(\sqrt{3^2} = 3\). This leaves us with \(3\sqrt{2}\).

• This helps in expressing the square root in a form that is often more straightforward to work with.
• Recognizing square numbers (like 9 in our example) is key to simplification.

In the solution \(x=\pm 3\sqrt{2}\), the expression is elegant and accurate, revealing how understanding the principle of simplifying radicals can transform a seemingly complex radical into an easier term to manage.