Understanding the Basics: Solving the 3x4 2 Equation

Introduction: In algebra, solving equations is a fundamental skill. In this article, we will explore the equation 3x4 2, a linear equation, and how to solve it. We'll also delve into the more complex quadratic equation 3x^2 4^2 9x^2 - 24x 16, and how to solve this using both factorization and the quadratic formula.

Solving the Linear Equation 3x 4 2

A linear equation involves a first-degree polynomial and has a single variable. To solve 3x 4 2, follow these steps:

Subtract 4 from both sides of the equation. Divide both sides by 3 to isolate x.

Let's solve it step by step:

3x 4 2 3x -2 (Subtract 4 from both sides) x -2/3 (Divide both sides by 3)

The solution is x -2/3.

Converting to Quadratic Form

Now, let's look at the quadratic equation 3x^2 4^2 9x^2 - 24x 16. We can use algebraic identities to rewrite this equation.

Using Algebraic Identities

The identity (a b)^2 a^2 2ab b^2 can be applied to both sides of the equation.

Applying this to both sides:

3x^2 4^2 9x^2 - 24x 16 (3x 4)^2 (3x - 4)^2

This means (3x 4)^2 (3x - 4)^2. Expanding and simplifying:

9x^2 24x 16 9x^2 - 24x 16 24x -24x 48x 0

Thus, x 0.

Solving the Quadratic Equation Using Factorization

Let's now solve the quadratic equation 9x^2 - 24x 16 0 using factorization.

First, expand the right-hand side using the identity (a - b)^2 a^2 - 2ab b^2:

(9x - 16)^2 0 9x - 16 0 9x 16 x 16/9

The solution is x 16/9.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form ax^2 bx c 0. The formula is:

x (-b ± √(b^2 - 4ac)) / 2a

For the equation 9x^2 - 24x 16 0:

a 9 b -24 c 16

First, calculate the discriminant:

D b^2 - 4ac (-24)^2 - 4 * 9 * 16 576 - 576 0

Since the discriminant is zero, there is exactly one solution:

x (-(-24) ± √0) / (2 * 9) 24 / 18 16 / 9

Hence, the solution is x 16/9.

Conclusion: In this article, we solved the simple linear equation 3x 4 2 and then tackled the more complex quadratic equation 3x^2 4^2 9x^2 - 24x 16. By using algebraic identities and the quadratic formula, we can solve a wide range of equations accurately.