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Explore the quadratic equation, its general form, methods for solving it, and an example calculation. Master algebra basics today!

Understanding the Quadratic Equation

The quadratic equation is a fundamental concept in algebra that appears in many different mathematical contexts. In this article, we will delve into the importance of the quadratic equation, its general form, and the methods used to solve it.

General Form of a Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form:

1. ax² + bx + c = 0

Here, ‘a’, ‘b’, and ‘c’ are constants, with ‘a’ ≠ 0, and ‘x’ is the variable. The quadratic equation represents a parabola when plotted on a graph. The constant ‘a’ determines the direction of the parabola (opening upwards if ‘a’ > 0, or downwards if ‘a’ < 0).

Methods of Solving Quadratic Equations

There are several methods to find the roots (solutions) of a quadratic equation. We will briefly discuss the three most common methods:

Factoring

Quadratic Formula

Completing the Square

Factoring

Factoring is the process of breaking down the quadratic equation into simpler terms that can be solved more easily. By expressing the equation as a product of two binomials, we can find the solutions by setting each binomial equal to zero and solving for ‘x’.

Quadratic Formula

The quadratic formula is a general solution to any quadratic equation. Derived from the process of completing the square, it provides a direct method for finding the roots of the equation:

1. x = (-b ± √(b² – 4ac)) / 2a

The quadratic formula takes into account the coefficients ‘a’, ‘b’, and ‘c’ to determine the solutions for ‘x’. It is particularly useful when factoring is not possible or cumbersome.

Completing the Square

Completing the square is a method that involves re-writing the quadratic equation in such a way that one side of the equation becomes a perfect square trinomial. This technique allows us to solve the equation by taking the square root of both sides and isolating ‘x’. It is particularly helpful in situations where factoring is difficult or when working with irrational roots.

Conclusion

Quadratic equations play a significant role in mathematics, with applications in diverse fields such as physics, engineering, and finance. Understanding the general form of the equation and the various methods to solve it are essential skills for anyone studying algebra or pursuing a career in a related field.

Example of Quadratic Equation Calculation

Let’s work through an example to demonstrate how to solve a quadratic equation using the quadratic formula. Consider the following equation:

1. 2x² – 5x + 2 = 0

To solve this equation using the quadratic formula, we identify the coefficients ‘a’, ‘b’, and ‘c’:

a = 2

b = -5

c = 2

Now, we can plug these values into the quadratic formula:

1. x = (-b ± √(b² – 4ac)) / 2a

Substituting the coefficients:

1. x = (-(-5) ± √((-5)² – 4 * 2 * 2)) / (2 * 2)

Simplifying the equation:

1. x = (5 ± √(25 – 16)) / 4

Continuing to simplify:

1. x = (5 ± √9) / 4

Since the square root of 9 is 3, we have two possible solutions for ‘x’:

x = (5 + 3) / 4 = 8 / 4 = 2

x = (5 – 3) / 4 = 2 / 4 = 1/2

Thus, the roots of the given quadratic equation are x = 2 and x = 1/2.