Problem 29 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$3 x^{2}=75$$

Step-by-Step Solution

Answer

The solutions are \(x = 5\) and \(x = -5\).

Step 1: Simplify the Equation

Start by simplifying the given equation. The equation is \(3x^2 = 75\). First, divide both sides by 3 to isolate \(x^2\) on one side. This gives \(x^2 = 25\).

Step 2: Take the Square Root

To solve for \(x\), take the square root of both sides of the equation. This gives \(x = \pm \sqrt{25}\), which simplifies to \(x = \pm 5\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is squared. They follow the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Solving these equations is fundamental in algebra because they appear in various practical scenarios, ranging from calculating areas to predicting demographic trends.
To adequately solve a quadratic equation, one can use different methods such as:

• Factoring
• Completing the square
• The quadratic formula

These methods assist in breaking down the equation to easily find the values of \(x\), which satisfy the equation. In our case, the equation \(3x^2 = 75\) is simplified to an equation of the form \(ax^2 = 0\), after simplifying.

Isolation of Variables

Isolation of variables is a critical step when working with algebraic equations. It refers to the process of rearranging an equation to get a particular variable by itself on one side of the equation, preferably the left side, making it easier to solve for that variable.

In our exercise, we start with the equation \(3x^2 = 75\). To isolate \(x^2\), we divide both sides of the equation by 3. This step simplifies the original equation to \(x^2 = 25\). By isolating \(x^2\), we make significant progress towards finding the solution for \(x\).

The main principle here is: whatever operation is performed on one side of the equation must be done on the other side to maintain equality. The goal is to steadily work towards simplifying the equation until you obtain a direct solution for the variable in question.

Applying the Square Root Method

After successfully isolating the variable \(x^2\), the next logical step is to apply the square root method. This technique is especially useful when the quadratic equation is in the form \(x^2 = n\), where \(n\) is a non-negative number. By taking the square root of both sides, you can find possible solutions for \(x\).

In the case of \(x^2 = 25\), taking the square root of each side gives \(x = \pm \sqrt{25}\). Simplifying \(\sqrt{25}\) gives \(x = \pm 5\).

• The positive root \(+5\) suggests one valid solution.
• The negative root \(-5\) indicates another potential solution.

When using the square root method, consider both the positive and negative roots. Quadratic equations usually have two solutions, due to the properties of squaring positive and negative numbers. This duality of solutions is crucial in ensuring all possible values of \(x\) are captured.