Problem 33 - 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. $$ x^{2}=16 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=16\) are \(x=4\) and \(x=-4\)

Step 1: Interpreting the Problem

This problem is asking for the solutions of the equation \(x^{2}=16\). The letter \(x\) stands for a number that we do not yet know.

Step 2: Applying the Square Root Property

The Square Root Property states that for any real number n and for any algebraic expression x, if \(x^{2}=n\), then \(x=\sqrt{n}\) or \(x=-\sqrt{n}\). By applying this rule, we can find the solutions of the equation. So we have: \(x=\sqrt{16}\) or \(x=-\sqrt{16}\).

Step 3: Calculating the Square Roots

We Calculate the square roots. We know that: \(\sqrt{16}=4\) and \(-\sqrt{16}=-4\). Hence, \(x=4\) or \(x=-4\).

Key Concepts

Square Root Property

The square root property is a powerful tool that simplifies solving quadratic equations of the form \(x^2 = n\). This property states that if \(x^2\) equals a number \(n\), then \(x\) must either be the positive square root of \(n\) or the negative square root of \(n\). In mathematical terms, \(x = \sqrt{n}\) or \(x = -\sqrt{n}\).
This property hinges on the fact that squaring a number always yields a positive result or zero, meaning any real number squared will be non-negative. Therefore, if \(x^2 = 16\), it tells us that \(x\) could either be \(4\) or \(-4\) because both numbers squared result in 16.
By leveraging the square root property, students can quickly solve these types of quadratic equations without needing to expand or factor expressions. It's like having a shortcut to the solution. Remember to always consider both the positive and negative roots as potential solutions to the equation.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and more. When solving quadratic equations, especially using the square root property, you'll often encounter square roots or radical expressions.
For example, in the equation \(x^2 = 16\), applying the square root property leaves us with the radical expression \(\sqrt{16}\). The term "radical" signifies the root sign itself. With our example, \(\sqrt{16} = 4\) and \( -\sqrt{16} = -4\).

• Understanding how to simplify radicals is crucial. For instance, \(\sqrt{16}\) simplifies to \(4\) because \(4 \times 4\) equals 16.
• Radical expressions are often left unsimplified when they do not result in whole numbers, giving rise to other forms, such as \(\sqrt{5}\) or \(-\sqrt{3}\).

Radical expressions are incredibly useful in mathematics because they allow you to express solutions precisely when they cannot be simplified to an integer.
Mastering these expressions can simplify calculation processes and provide exact answers, crucial for advanced math problems.

Real Number Solutions

In mathematics, real number solutions refer to solutions of equations that are real numbers, not imaginary or complex. Real numbers include rational numbers, like integers and fractions, and irrational numbers. When solving equations like \(x^2 = 16\), we're often interested in finding real number solutions.
Real numbers are significant because they correspond to actual quantities in the real world, unlike complex numbers which are used in theoretical contexts. In our example, the solutions \(x = 4\) or \(x = -4\) are real numbers because they represent points on the number line.

• Sometimes equations might not have real number solutions at all, such as \(x^2 = -1\), in which case the solutions would be complex numbers.
• Understanding whether solutions are real helps determine their applicability and relevance in physical problems.

Emphasizing real number solutions is critical in fields ranging from engineering to economics, where practical, tangible results are necessary. Thus, identifying whether a solution is real or not ensures clarity and practicality in problem-solving scenarios.