Problem 98 - Algebra and Trigonometry

Question

Solve equation by the method of your choice. $$ 3 x^{2}-27=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(3x^{2} - 27 = 0\) are \(x = -3\) and \(x = 3\).

Step 1: Rearrange the equation

First, let's rearrange the equation in order to make it easier to solve. We can start by adding 27 to both sides to get the equation: \(3x^{2} = 27\).

Step 2: Solve for \(x^{2}\)

Next, we can divide both sides by 3 to isolate \(x^{2}\). This gives us the equation: \(x^{2} = 9\).

Step 3: Solve for \(x\)

To find \(x\), we take the square root of both sides. This gives us two solutions since the square root of a number can be both positive and negative. Therefore, the solutions to the equation are \(x = -3\) and \(x = 3\).

Key Concepts

Square Root Method

When solving quadratic equations, a variety of methods can be employed, and one of the simplest approaches is the Square Root Method. This method is especially useful when the quadratic equation is already in the form of \(ax^2 = c\). Here’s how it works:

• First, isolate the \(x^2\) term by moving all other terms to the other side of the equation. This step makes it easier to take the square root later.
• Once the equation is in the form \(x^2 = c\), the next step is to apply the square root to both sides of the equation.
• Remember that whenever we take the square root of an equation, there are two potential solutions: one positive and one negative. This is due to the properties of squares in algebra. For example, both \(3^2\) and \((-3)^2\) equal 9.

This method is straightforward and effective, though it only works directly when the quadratic equation is suited to it, typically when there is no linear \(x\) term present.

Solving Equations

Solving equations is a central aspect of algebra and involves finding the value or values of variables that make the equation true. Here’s a quick guide to understanding this process:

• Identify your equation type: Linear, quadratic, or others. In this instance, we focus on quadratic equations.
• Choose a method based on the equation type and form. For a simple quadratic equation like \(3x^2 = 27\), isolating \(x^2\) and applying the square root method is effective.
• Perform algebraic operations step-by-step, such as addition, subtraction, multiplication, or division, to rearrange the equation.
• Check your solutions by substituting them back into the original equation to ensure they satisfy the equation’s conditions.

Applying these steps systematically allows for solving equations efficiently and confidently.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging, simplifying, or rewriting equations to make them more manageable or solve them. Here’s how it works:

• Start by identifying terms and operators within the equation. Here, in \(3x^2 - 27 = 0\), the goal is to simplify or isolate the terms.
• Use arithmetic inversion, like adding or subtracting values across the equation, to move terms and simplify the equation structurally. For example, adding 27 to both sides to get \(3x^2 = 27\).
• Apply multiplication or division to further simplify. Dividing both sides by 3 simplifies the equation to \(x^2 = 9\).
• Throughout the process, maintain balance by performing the same operations on both sides of the equation, ensuring equality is preserved.

This approach allows for solving more complex problems by breaking them down into simpler, manageable steps, facilitating easier problem-solving in algebra.