Problem 29 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ n^{2}=49 $$

Step-by-Step Solution

Answer

The solutions of the equation \(n^{2}=49\) are \(n = 7\) and \(n = -7\)

Step 1: Identify the problem

The problem given is an equation where n squared equals 49, written as \(n^{2} = 49\). The goal is to find the possible values of n that satisfy this equation.

Step 2: Use the square root principle

To find n, we must take the square root of both sides of the equation. However, remember that the square root of a number y yields positive and negative solutions. So for \(n^{2} = 49\), we apply the square root giving us \(n = \sqrt{49}\) and \(n = -\sqrt{49}\)

Step 3: Simplify the radical expressions

\The square root of 49 is 7 (since \(7 * 7 = 49\). Therefore, the two solutions of the equation are \(n = 7\) and \(n = -7\)

Key Concepts

Square Root Principle

The square root principle is a fundamental concept in algebra, particularly when it comes to solving quadratic equations like the one in our exercise, where you have an equation of the form \(x^2 = a\). This principle states that if \(x^2 = a\), then \(x\) can be either the positive square root of \(a\) or the negative square root of \(a\).

To put it into practice, let's say we have the equation \(n^2 = 49\). According to the square root principle, we will take the square root of both sides of the equation, remembering to include both the positive and negative possibilities. That means \(n\) can be \(+\sqrt{49}\) or \(-\sqrt{49}\). By squaring both 7 and -7, we find that they both satisfy the original equation \(n^2 = 49\), demonstrating the square root principle effectively.

Radical Expressions

A radical expression is an expression that includes a radical symbol (\(\sqrt{}\)), which indicates the square root or higher roots of a number. In the context of solving quadratic equations, once we apply the square root principle, what we often get are radical expressions.

For instance, after applying the square root principle to the equation \(n^2 = 49\), we ended up with \(n = \sqrt{49}\) and \(n = -\sqrt{49}\). The \(\sqrt{49}\) here is a radical expression. To simplify, we evaluate the expression under the radical, which in this case is a perfect square. Thus, \(\sqrt{49}\) simplifies to 7, and the equation \(n = \sqrt{49}\) simplifies to \(n = 7\), making the expression an integer. Always check if the expression under the radical can be simplified before writing down your final answer.

No Real Solution

There are scenarios where a quadratic equation has no real solution. This happens when we have a negative number inside the square root, known as a 'negative radicand.' Since the square root of a negative number is not a real number, equations that would require such a square root have no real solutions in the set of real numbers.

For example, if our equation was \(n^2 = -49\), taking the square root would lead to \(n = \sqrt{-49}\), which does not exist in the real number system. In such cases, the answer to the equation is that there is no real solution. However, it's worth mentioning that in the field of complex numbers, such roots are defined and lead to what we call imaginary numbers. But for real-number equations, we stick to stating there is no real solution when encountering a negative radicand.