Problem 77 - Elementary Algebra

Question

Solve. $$ 3 x 2-12=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -2\).

Step 1: Understand the Equation

The given equation is \(3x^2 - 12 = 0\). This is a quadratic equation in standard form, \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = 0\), and \(c = -12\).

Step 2: Move Constant Term to the Other Side

Add 12 to both sides of the equation to isolate the quadratic term: \(3x^2 = 12\).

Step 3: Simplify the Coefficient of the Quadratic Term

Divide both sides of the equation by 3 to simplify: \(x^2 = 4\).

Step 4: Solve for x Using Square Roots

Take the square root of both sides to solve for \(x\): \(x = \pm \sqrt{4}\).

Step 5: Simplify the Square Roots

Simplify \(\sqrt{4}\) to get \(x = \pm 2\).

Key Concepts

Solving Quadratic Equations

Quadratic equations play a key role in algebra and can appear in various forms. Understanding how to solve them is essential for grasping more advanced mathematical concepts. A quadratic equation typically appears in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the unknown variable.
In the example we are discussing, the equation is \(3x^2 - 12 = 0\), which breaks down to \(a = 3\), \(b = 0\), and \(c = -12\).
The key to solving a quadratic equation is to first aim to isolate the \(x^2\) term on one side of the equation. This often involves moving other terms across the equation sign, either by addition, subtraction, multiplication, or division. Once the \(x^2\) term is isolated, the equation can be solved using methods such as factoring, completing the square, or applying the quadratic formula, with the decision on the method largely depending on the form of the equation.

Square Roots

After isolating the \(x^2\) term in a quadratic equation, the next step is often to utilize square roots. In our example, after simplifying the equation \(3x^2 = 12\) to \(x^2 = 4\), we take the square root of each side. The concept of square roots here introduces the principle that \(x\) can have both positive and negative roots.
Taking the square root of \(x^2 = 4\) yields \(x = \pm \sqrt{4}\). This is a critical step in the problem-solving process because it acknowledges two potential solutions for \(x\). It’s essential to remember that square roots can result in a dual outcome: one positive and one negative, since multiplying two negative numbers also gives a positive product.
Therefore, simplifying \(\sqrt{4}\), we find \(x = 2\) or \(x = -2\). These methods make it essential to understand that solving \(x^2 = n\) involves both \(x = \sqrt{n}\) and \(x = -\sqrt{n}\).

Algebraic Equations

Algebraic equations are central to mathematical problem-solving and include any equations that involve algebraic expressions. These can range from linear to quadratic and beyond. When dealing with quadratic equations, recognizing and manipulating complex algebraic expressions is crucial.
In the equation \(3x^2 - 12 = 0\), we recognize this as a quadratic equation due to the \(x^2\) term, which signifies a degree of 2. Solving such equations often involves a series of algebraic manipulations:

• Relocating terms, such as moving \(-12\) to the other side of the equation.
• Changing coefficients, like dividing through by 3 to simplify \(3x^2 = 12\).
• Employing algebraic identities and rules, especially when simplifying square roots.

Mastery of these basic operations is key to solving not just quadratics, but a wide array of algebraic problems. Understanding how each manipulation impacts the equation's balance is fundamental to real-world applications and more advanced mathematical studies.