Solving Quadratic Equations: Methods and Solutions

Quadratic equations are a fundamental concept in algebra, often represented in the form (ax^2 bx c 0). These equations can be solved using various methods, including factoring, completing the square, and the quadratic formula. In this article, we will explore these methods and provide examples to illustrate how to solve quadratic equations.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, with the general form (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). The solutions to these equations can be found using different methods, each with its own advantages and applications.

Solving Quadratic Equations: Methods

1. Factoring

Factoring is one of the most straightforward methods for solving quadratic equations. This method involves breaking down the quadratic expression into a product of two linear factors.

Example:

1. Solve the equation (x^2 - 4x 3 0)

Step 1: Factor the quadratic expression.
[x^2 - 4x 3 (x - 3)(x - 1) 0]

Step 2: Set each factor to zero and solve for (x).

[x - 3 0 implies x 3]
[x - 1 0 implies x 1]

2. Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. It is given by:

[x_{12} frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Example:

2. Solve the equation (2x^2 - 5x - 3 0)

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

[a 2,; b -5,; c -3]

Step 2: Substitute these values into the quadratic formula.

[x frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 2 cdot (-3)}}{2 cdot 2}]

Step 3: Simplify the expression.

[x frac{5 pm sqrt{25 24}}{4} frac{5 pm sqrt{49}}{4} frac{5 pm 7}{4}]

Step 4: Solve for (x).

[x_1 frac{5 7}{4} frac{12}{4} 3]
[x_2 frac{5 - 7}{4} frac{-2}{4} -frac{1}{2}]

Therefore, the solutions are (x 3) and (x -frac{1}{2}).

3. Completing the Square

Completing the square is another method for solving quadratic equations. This technique involves transforming the equation into a perfect square trinomial, which can then be easily solved.

Example:

3. Solve the equation (x^2 - 4x 3 0)

Step 1: Move the constant term to the right side of the equation.

[x^2 - 4x -3]

Step 2: Add the square of half the coefficient of (x) to both sides of the equation.

[left(frac{-4}{2}right)^2 4]
[x^2 - 4x 4 -3 4]
[x^2 - 4x 4 1]

Step 3: Rewrite the left side as a perfect square.

[(x - 2)^2 1]

Step 4: Take the square root of both sides.

[x - 2 pm 1]

Step 5: Solve for (x).

[x 2 1 3]
[x 2 - 1 1]

Therefore, the solutions are (x 3) and (x 1).

Conclusion

Quadratic equations are a common type of algebraic equation that can be solved using several methods. Factoring, the quadratic formula, and completing the square are some of the most popular and effective methods. Each method has its own advantages and can be chosen based on the specific problem and the values of the coefficients.