Problem 87 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 3 x^{2}=60 $$

Step-by-Step Solution

Answer

The solutions to the equation \(3x^2 = 60\) are \(x = 2\sqrt{5}\) and \(x = -2\sqrt{5}\)

Step 1: Isolate the square term

Let's start off by isolating the square term, \(x^2\). This can be done by dividing the whole equation by 3. That would give us the simplified equation: \(x^2=60/3\) or \(x^2=20\)

Step 2: Solve for \(x\)

Once the square term has been isolated, it's possible solve for \(x\), simply by square rooting both sides. Remember that square rooting gives a positive and a negative value. This means \(x = \sqrt{20}\) and \(x = -\sqrt{20}\)

Step 3: Simplify

\(\sqrt{20}\) can be simplified to \(2\sqrt{5}\). Hence, the solutions for the equation are \(x = 2\sqrt{5}\) and \(x = -2\sqrt{5}\)

Key Concepts

Isolation of Square Term

The first step in solving a quadratic equation like the one given, involves isolating the square term. This means we want to get the variable squared, such as \(x^2\), by itself on one side of the equation. In the given problem, we start with \(3x^2 = 60\).

To do this, we divide both sides of the equation by 3, which is the coefficient of \(x^2\). This operation helps simplify the expression and separate the squared variable from any other numbers attached to it by multiplication or division. Doing the division, we get \(x^2 = 20\). Now, the square term \(x^2\) is neatly isolated, making it easier to solve the equation in the next step.

Square Rooting Method

Once the square term is isolated, we can proceed by applying the square rooting method to solve for \(x\). This method involves taking the square root of both sides of the equation. Be cautious: taking the square root of a number produces both a positive and a negative value.

For this equation, we have \(x^2 = 20\). By taking the square root of both sides, we get \(x = \sqrt{20}\) and \(x = -\sqrt{20}\). Remember, both solutions are valid for the equation since \((-x)^2\) is always equal to \(x^2\).

This technique capitalizes on the fact that squaring a number (positive or negative) results in a positive outcome. Always remember to account for both the positive and negative solutions when employing the square rooting method.

Simplification of Radicals

After finding \(x = \sqrt{20}\) and \(x = -\sqrt{20}\), the next step involves the simplification of radicals. Simplifying radicals means breaking down a square root into its simplest form, when possible.

The number 20 can be factored into 4 and 5, where 4 is a perfect square. Thus, \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5}\). Since \(\sqrt{4} = 2\), we can therefore simplify \(\sqrt{20}\) to \(2\sqrt{5}\).

This gives the final answers for \(x\) as \(x = 2\sqrt{5}\) and \(x = -2\sqrt{5}\). Simplifying radicals not only provides a cleaner solution, but also prepares the equation for any further mathematical use, ensuring calculations are as accurate and simplified as possible.