Problem 51 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}=49\)

Step-by-Step Solution

Answer

The real solutions are \(x = 7\) and \(x = -7\).

Step 1: Setting Up the Equation

The equation provided is \(x^{2} = 49\). This is a quadratic equation that we need to solve for \(x\). The goal is to find all real solutions that satisfy this equation.

Step 2: Applying the Square Root Method

To solve \(x^{2} = 49\), take the square root of both sides of the equation. This gives us two potential solutions because both \((+7)^{2}\) and \((-7)^{2}\) equal 49. Therefore, \(x = 7\) and \(x = -7\). This step involves recognizing that the square root operation can yield both positive and negative results.

Step 3: Checking the Solutions

Substitute the found solutions back into the original equation to verify their accuracy. For \(x = 7\), \(7^{2} = 49\), which is true. For \(x = -7\), \((-7)^{2} = 49\), which is also true. Both solutions satisfy the equation.

Key Concepts

Understanding the Square Root Method

The square root method is a straightforward way to solve simple quadratic equations of the form \(x^2 = a\). The method involves taking the square root on both sides of the equation, which simplifies the expression. When you apply the square root method, remember there are always two possible roots — a positive and a negative root.

• To apply the method, identify the number on the right side of the equation, such as 49 in this case.
• Calculate the square root of this number. For 49, the principal square root is 7, as \(7^2 = 49\), but don’t forget the other root, -7, because \((-7)^2 = 49\) as well.

This method is effective in cases where the equation is neatly set as \(x^2 = ...\). Always verify your solutions by substituting them back into the original equation to ensure they satisfy it.

Identifying Real Solutions

Real solutions are values of \(x\) that satisfy a given equation and are considered 'real' numbers. Real numbers include all positive and negative integers, fractions, and decimals, but exclude imaginary numbers. In the context of quadratic equations like \(x^2 = 49\), finding real solutions means identifying values of \(x\) that make the equation true without involving imaginary numbers.

• Real solutions can be found directly from the square root method, yielding \(x = 7\) and \(x = -7\).
• Both these solutions are real numbers and make \(x^2 = 49\) true.
• Real solutions are crucial as they often represent practical and applicable results in real-world scenarios.

In this simple quadratic equation, both 7 and -7 are real numbers that make the equation work.

Analyzing the Power of a Variable

The term "power of a variable" refers to the exponent in an equation. In \(x^2 = 49\), the power of the variable \(x\) is 2. The power determines the number of solutions an equation might have. For any equation with the variable raised to the power of 2, known as a quadratic equation, you can anticipate up to two potential solutions.

• The power signifies that when solving, you should prepare for two potential values of \(x\) that satisfy the equation, reflecting the dual nature of squaring a positive and negative value.
• Understanding this concept helps in judging the nature of the solutions, considering whether they are real or complex.
• In simple terms, when the highest power of \(x\) is 2, it brings about a parabolic graph indicating two intersections with the \(x\)-axis, thereby two solutions.

Understanding the power of a variable in equations is pivotal to solving and predicting the number of possible solutions effectively.