Solving Quadratic Equations: A Comprehensive Guide to Solving 3x2 - 12x 9 0

Quadratic equations are a fundamental part of algebra and play a crucial role in many areas of mathematics and its applications. In this article, we will explore how to solve a specific quadratic equation: 3x2 - 12x 9 0. We will look at multiple methods to solve this equation, ensuring that readers understand the process and rationale behind each step.

Introduction to Quadratic Equations

A quadratic equation is an equation of the form ax2 bx c 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation are the x-values that satisfy the equation, often referred to as the roots or zeros of the quadratic equation.

Solving 3x2 - 12x 9 0 Using Factoring

Let's start by solving the equation using the factoring method.

Step 1: Simplify the equation if possible. In this case, the equation is already in a simplified form.

Step 2: Factor the quadratic expression. We need to find two numbers that multiply to 3 * 9 27 and add up to -12. These numbers are -3 and -9.

Step 3: Rewrite the equation as a product of its factors.

3x2 - 12x 9 0 →

3(x2 - 4x 3) 0

x2 - 4x 3 (x - 3)(x - 1) 0

Step 4: Set each factor equal to zero and solve for x.

x - 3 0 → x 3

x - 1 0 → x 1

Therefore, the solutions are x 3 and x 1.

Solving 3x2 - 12x 9 0 Using the Quadratic Formula

The quadratic formula provides a general method for solving any quadratic equation ax2 bx c 0. The formula is given by:

x [-b ± sqrt(b2 - 4ac)] / (2a)

Step 1: Identify the coefficients a, b, and c.

a 3, b -12, c 9

Step 2: Substitute the values into the quadratic formula.

x [12 ± sqrt(144 - 108)] / 6

x [12 ± sqrt(36)] / 6

x [12 ± 6] / 6

Step 3: Solve for the two possible values of x.

x (12 6) / 6 18 / 6 3

x (12 - 6) / 6 6 / 6 1

The solutions are x 3 and x 1.

Solving 3x2 - 12x 9 0 Using Completing the Square Method

This method involves transforming the quadratic equation into a perfect square trinomial.

Step 1: Ensure the coefficient of x2 is 1. If not, divide the entire equation by the coefficient of x2.

x2 - 4x 3 0

Step 2: Move the constant term to the other side.

x2 - 4x -3

Step 3: Add the square of half the coefficient of x to both sides.

x2 - 4x 4 -3 4

(x - 2)2 1

Step 4: Take the square root of both sides.

x - 2 ±1

Step 5: Solve for x.

x 2 ± 1

Therefore, the solutions are x 3 and x 1.

Conclusion

In conclusion, the solutions to the quadratic equation 3x2 - 12x 9 0 are x 3 and x 1. Whether you choose to factor, use the quadratic formula, or complete the square, the process involves breaking down the problem into more manageable steps. Understanding these methods can greatly enhance your problem-solving skills in algebra.

Keyword: quadratic equation, solving equations, algebraic solutions