Solving Quadratic Equations: A Step-by-Step Guide for Beginners

Quadratic equations are a fundamental part of algebra, and they can appear in a variety of mathematical and real-world applications. If you're new to this topic, you might wonder, what is x if 4x^2 8x - 12 0? This article will walk you through the process of solving such equations, providing clear examples and explanations.

Understanding Quadratic Equations

A quadratic equation is an equation of the form ax^2 bx c 0, where a, b, and c are constants, and a is not equal to 0. The general form of the quadratic equation given is 4x^2 8x - 12 0. This particular equation can be solved to find the values of x that satisfy it.

Step 1: Simplify the Equation

Let's start by simplifying the given equation, 4x^2 8x - 12 0. We can divide the entire equation by 4 to make it simpler.

Divide each term by 4: 4x^2/4 8x/4 - 12/4 0 which simplifies to: x^2 2x - 3 0

Step 2: Factor the Equation

Next, we need to factor the quadratic equation. This can often be done by finding two numbers that multiply to give the constant term (-3) and add up to the coefficient of the linear term (2).

x^2 2x - 3 0 x^2 3x - x - 3 0 x(x 3) - 1(x 3) 0 (x - 1)(x 3) 0

Now, we have the equation in factored form, (x - 1)(x 3) 0.

Step 3: Apply the Zero Product Rule

The zero product rule states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x.

x - 1 0 or x 3 0

x - 1 0 implies x 1 x 3 0 implies x -3

Thus, the solutions to the equation are x 1 and x -3.

Alternative Method: Completing the Square

Another way to solve the quadratic equation is by completing the square.

4x^2 8x - 12 0 Add 12 to both sides: 4x^2 8x 12 x^2 2x 3 Complete the square by adding (2/2)^2 1 to both sides: x^2 2x 1 3 1 which simplifies to: (x 1)^2 4 Take the square root of both sides: x 1 ±2 So, x -1 ± 2 which gives x 1 or x -3.

Conclusion

Quadratic equations can be solved using various methods, including factoring and completing the square. By following these steps, you can find the values of x that satisfy the equation. Understanding these methods is essential for solving more complex algebraic problems in mathematics and real-world applications.

Keywords

quadratic equations

Quadratic equations are polynomial equations of degree 2. They are expressed in the form ax^2 bx c 0, where a, b, and c are coefficients, and a ≠ 0. These equations are used to describe a variety of phenomena in science, engineering, and finance.

roots of quadratic equations

The roots of a quadratic equation are the values of x that make the equation true. They can be found using factoring, completing the square, or the quadratic formula. The roots can be real or complex numbers, depending on the properties of the equation.