Solving Quadratic Equations: Techniques and Examples

Quadratic equations are fundamental in algebra and are widely used in mathematics and its applications. A quadratic equation can be solved using several techniques, including the quadratic formula, derivates parcial, factorization, and other methods. This article will explore these techniques and provide a detailed solution to a specific example.

Introduction to Quadratic Equations

A quadratic equation is an equation of the form ax2 bx c 0, where a, b, c are constants and a ≠ 0. The solutions to these equations can be found using various methods.

Quadratic Formula Method

The most common and straightforward method to solve a quadratic equation is by using the quadratic formula. The formula is given by:

x [-b ± √(b2 - 4ac)] / 2a

Here, a is the coefficient of x2, b is the coefficient of x, and c is the constant term.

Example: Solving the Equation y2 - 4y 4 0

Let's solve the equation y2 - 4y 4 0 using the quadratic formula.

Step 1: Identify the coefficients

From the equation y2 - 4y 4 0, we have: a 1, b -4, c 4

Step 2: Calculate the discriminant

The discriminant, Δ b2 - 4ac, is calculated as:

Δ 16 - 16 0

Step 3: Substitute the values into the quadratic formula

Now, substitute the values into the quadratic formula:

y [-(-4) ± √(0)] / (2 * 1)

y [4 ± 0] / 2

y 4 / 2 2

Hence, the solution to the equation is y 2.

Derivatives Parcial Method

Another method to solve a quadratic equation is by using derivatives. However, it's important to note that this method is not as commonly used as the quadratic formula.

For the equation y2 - 4y 4 0, we can differentiate and solve for the critical points:

Given: f(y) y2 - 4y 4

First derivative: f'(y) 2y - 4

Solving for critical points: 2y - 4 0 implies y 2

Factorization Method

The factorization method involves breaking down the quadratic equation into simpler expressions. For the equation y2 - 4y 4 0 we can factorize it as follows:

y2 - 4y 4 (y - 2)(y - 2)

Setting each factor to zero gives the solution: y 2

Conclusion

In conclusion, solving quadratic equations can be done through various methods including the quadratic formula, derivatives parcial, factorization, and other techniques. Each method has its own advantages and can be used depending on the specific requirements of the problem at hand. The quadratic formula, in particular, is a powerful tool that provides a straightforward and reliable solution to quadratic equations.