Problem 27 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible. $$3 x^{2}=60$$

Step-by-Step Solution

Answer

Therefore, the solutions are \(x_{1} = 2 \sqrt{5}\) and \(x_{2} = -2 \sqrt{5}\).

Step 1: Isolate the term with the variable

Divide both sides of the equation by 3 to isolate \(x^2\). This gives us: \[x^2 = 60/3 = 20.\]

Step 2: Solve for x

Take the square root of both sides of the equation to solve for x. As the square root of a square is the absolute value of the number, when we take the square root of both sides we get two solutions (positive and negative). This gives us: \[x_{1,2} = \pm \sqrt{20}.\]

Step 3: Simplify

\(\sqrt{20}\) can be simplified further by factoring out the perfect square of \(20 = 4 \times 5\). Therefore, \( \sqrt{20} = \sqrt{4 \times 5} = 2 \sqrt{5} \). So we have two solutions: \[x_{1} = 2 \sqrt{5},\] and \[x_{2} = -2 \sqrt{5}.\]

Key Concepts

Simplifying Radicals

Simplifying radicals refers to the process of making a square root expression as simple as possible. To do this effectively, you need to look for perfect squares in the number under the square root. A perfect square is a number that can be expressed as the square of an integer. For instance, 4 is a perfect square because it is equal to 2 squared.

Consider the problem, where we were tasked with simplifying \( \sqrt{20} \). To simplify \( \sqrt{20} \), we need to express 20 as a product of its factors. We find that 20 can be written as \( 4 \times 5 \). Since 4 is a perfect square, we can take the square root of 4, which is 2, outside of the radical symbol.

• Under the square root: \( 4 \times 5 \)
• Take the square root of the perfect square 4: \( \sqrt{4} = 2 \)
• Remaining inside the square root: 5

The simplified form of \( \sqrt{20} \) is \( 2\sqrt{5} \). This is a cleaner, more manageable expression.

Factoring

Factoring is the process of breaking down numbers or expressions into their multipliers or components. This is useful in simplifying complex mathematical expressions, including radicals or quadratic equations. In the context of our original problem, it helped in simplifying the radical expression during the final stage of solving the equation.

To factor a number, you determine the set of smaller numbers that multiply together to form the original number. Consider number 20 as used in simplifying \( \sqrt{20} \):

• Identify factors of 20: 1, 2, 4, 5, 10, 20
• Select the pair of factors \( 4 \times 5 \)

Factoring is important because it allows us to manipulate equations in a way that simplifies the solving process. Always consider all possible factor pairs, especially when solving quadratic equations, since factoring into perfect squares can greatly simplify the radicals.

Square Root Method

The square root method is a technique used to solve quadratic equations that can be expressed in the form \( ax^2 + c = 0 \). The goal is first to isolate the term \( x^2 \) and then use the property that the square and the square root operations are inverse to each other.

Start by isolating \( x^2 \), as was done in the equation \( 3x^2 = 60 \). Dividing both sides by 3 gives:
\[ x^2 = 20 \]
Next, apply the square root method by taking the square root of both sides:
\[ x = \pm \sqrt{20} \]

• This results in two possible values for x, one positive and one negative since squaring either will return the original value under the square root.

In this example, we further simplified \( \sqrt{20} \) to \( 2\sqrt{5} \), resulting in the solutions \( x = 2\sqrt{5} \) and \( x = -2\sqrt{5} \). The square root method is particularly advantageous for simple equations, providing quick and intuitive solutions when the quadratic is easily rewritten with \( x^2 \) isolated.