Problem 54 - Algebra 1

Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$x^{2}=36$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=36\) are \(x=6\) and \(x=-6\)

Step 1: Identify the equation

The equation we need to solve is \(x^{2}=36\)

Step 2: Apply square root to both sides

Take the square root of both sides of the equation to solve for \(x\). This eliminates the square on the left side and provides a solution (two possible values) on the right. This operation should be performed considering both the positive and negative results because squaring either a positive or a negative number results in a positive outcome. Hence, the solution to the equation is \(x= \pm \sqrt{36}\)

Step 3: Simplify the square root

\(\sqrt{36}\) simplifies to 6, because \(6^{2}=36\). Hence, the solutions for the equation are \(x= \pm 6\)

Key Concepts

Square Root Method

The square root method is a straightforward technique used to solve quadratic equations like \(x^2=36\). This approach is particularly effective when the quadratic equation can be written in the form \(x^2 = k\), where \(k\) is a constant. Here's how it works:

• Identify the Equation: Start with a quadratic equation such as \(x^2 = 36\).
• Apply the Square Root: To solve for \(x\), take the square root of both sides. Remember to consider both the positive and the negative roots. The square root of each side is \(x = \pm \sqrt{k}\).
• Solve: Simplify the square root to get your solution. For \(x^2 = 36\), this results in \(x = \pm 6\).

By applying this method, you systematically eliminate the square by counteracting it with a square root. It's important to note that squaring a positive or negative number yields the same positive result. Therefore, always consider both potential values to ensure you don't miss any solutions.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In the context of solving equations, radical expressions come into play when you express solutions using a root symbol, particularly when a number does not simplify to an integer.

• Definition: A radical expression is an expression that includes a root symbol (e.g., \( \sqrt{} \), \( \sqrt[3]{} \)).
• Simplification: If a radical expression can be simplified to an integer, it's beneficial to do so. In the case of \(\sqrt{36}\), it simplifies neatly to 6.
• General Usage: When dealing with radicals under the square root method, you aim to either simplify them to integers or present them in their simplest radical form if they cannot be simplifed further.

Learning how to manipulate radical expressions is crucial because they frequently appear in solutions. Understanding the simplification process assists in recognizing equivalency between radicals and whole numbers.

Integer Solutions

Integer solutions are solutions to equations that can be expressed as whole numbers, without any fractions or decimals. When solving quadratic equations like \(x^2 = 36\), identifying integer solutions is often the goal, as they tend to be simpler and more intuitive.

• Whole Numbers: An integer solution means both positive and negative whole numbers are considered. Here, the integer solutions are \(x = 6\) and \(x = -6\).
• Verification: To confirm these solutions are correct, you can substitute them back into the original equation \(x^2 = 36\) to check if they satisfy the equality.
• Simplicity: Opting for integer solutions when possible is often preferred over radical expressions when they simplify neatly.

Recognizing integer solutions not only ensures comprehension but also streamlines calculations. Embracing this approach allows for a clearer understanding of the balance between equations and their solutions.