Problem 2 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$x^{2}=100$$

Step-by-Step Solution

Answer

The solutions of the equation \( x^{2} = 100 \) are \( x = 10 \) and \( x = -10 \).

Step 1: Identify the square and the number

The equation is \( x^{2} = 100 \). Here, \( x^{2} \) is the square and 100 is the number to which the square is equal.

Step 2: Apply the square root property

Once we have identified the square and the number, we apply the square root property. This states that if \( x^{2} = a \), then \( x = \sqrt{a} \) or \( x = -\sqrt{a} \). Applying this to our equation gives \( x = \sqrt{100} \) or \( x = -\sqrt{100} \)

Step 3: Simplify the square root

The next step is to simplify the square root. The square root of 100 is 10. Therefore, \( x = 10 \) or \( x = -10 \)

Step 4: Check the solutions

The final step is to check the solutions. This involves substituting the solutions back into the original equation and seeing if they hold true. Substituting \( x = 10 \) back into \( x^{2} = 100 \) gives \( 10^{2} = 100 \), which is true. Similarly, substituting \( x = -10 \) back into \( x^{2} = 100 \) gives \( (-10)^{2} = 100 \), which is also true. Therefore, both \( x = 10 \) and \( x = -10 \) are solutions of the equation.

Key Concepts

Square Root Property

One of the key tools to solve quadratic equations of the form \( x^2 = a \) is the square root property. This property simplifies the process by allowing us to take the square root of both sides of the equation. It states that if \( x^2 = a \), then \( x = \sqrt{a} \) or \( x = -\sqrt{a} \).
This is because squaring a positive or a negative number results in the same positive number. Thus, both the positive and negative roots need to be considered as solutions.

• Step 1: Identify the squared term (e.g., \( x^2 \)) and the number it equals (e.g., \( 100 \)).
• Step 2: Apply the square root property by setting \( x = \sqrt{100} \) and \( x = -\sqrt{100} \).

By following these steps, you can solve any quadratic equation given in this particular format using the square root property.

Solving Equations

When solving equations, particularly quadratics, it's essential to understand the process involved. Equations like \( x^2 = 100 \) require specific strategies to find their solutions efficiently.
The goal of solving equations is to find all possible values of the variable that make the equation true.

• Equating the equation's sides and simplifying if possible.
• Considering both positive and negative values when taking square roots, as discussed in the square root property.
• Always verify your solutions by substituting back into the original equation to ensure they satisfy the equation.

Incorporating these steps helps verify accuracy and correctness, ensuring no solutions are overlooked.

Simplifying Radicals

Simplifying radicals is often necessary to arrive at the simplest form of your solutions. A radical expression like \( \sqrt{100} \) can often be simplified to a whole number. Here's how to simplify radicals:

• Factor the number inside the radical to prime factors if it isn't a perfect square.
• Identify and take out any pairs since the square root of a perfect square returns a non-radical whole number.

For example, \( \sqrt{100} = 10 \) because 100 is a perfect square of 10. Simplifying radicals not only makes the solution clearer, but also more manageable and easier to verify.