Solving Quadratic Equations: A Comprehensive Guide

Hey everyone! Let's dive into the world of quadratic equations. We're going to tackle the equation 13 + 2x² = 10x step by step. Solving quadratic equations might seem a bit daunting at first, but trust me, with a little practice and understanding, you'll be acing these problems in no time. In this comprehensive guide, we will explore different methods to solve quadratic equations, ensuring that you have a solid grasp of the concepts. We'll start with the basics, like understanding what a quadratic equation actually is, and then move on to the different techniques you can use to find the solutions.

Understanding Quadratic Equations

Before we jump into solving the equation 13 + 2x² = 10x, let's quickly recap what a quadratic equation is. A quadratic equation is an equation that can be written in the standard form: ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The 'x' represents the unknown variable that we're trying to solve for, and the exponents (the little '2' in x²) tells us it's a quadratic equation. The graph of a quadratic equation is a parabola, a U-shaped curve. The solutions to a quadratic equation, also known as the roots or zeros, are the x-values where the parabola intersects the x-axis.

When we look at our equation, 13 + 2x² = 10x, it might not look like the standard form yet. Our first step is to rearrange it so that it does. We want all the terms on one side of the equation and zero on the other side. That way, we can clearly see what the a, b, and c values are and use our solving techniques more easily. Remember, the a, b, and c values are critical for determining the solutions of the quadratic equation. The solutions can be real numbers, complex numbers, or even repeated roots. This depends on the values of a, b, and c in the equation and how they relate to each other. When you fully grasp the concept of quadratic equations, you will realize that it is not just about finding the value of 'x'; it is also about understanding the nature and behavior of the quadratic equation. It will also make you ready to solve the equation in our first example 13 + 2x² = 10x.

Rearranging the Equation

Let's rearrange 13 + 2x² = 10x to the standard form. First, we subtract 10x from both sides to get all terms on one side: 2x² - 10x + 13 = 0. Now, we can clearly see that a = 2, b = -10, and c = 13. The standard form allows us to use the quadratic formula, completing the square, or factoring to find the values of x that satisfy the equation. Also, we can determine the nature of the roots. Now that the equation is in standard form, we can determine if the equation will have two real solutions, one real solution, or no real solutions.

Methods for Solving Quadratic Equations

There are several methods we can use to solve quadratic equations. The three main methods are factoring, completing the square, and using the quadratic formula. We'll explore each of these, starting with the quadratic formula. Remember, choosing the best method depends on the specific equation. Some equations factor easily, while others might be best solved using the quadratic formula or completing the square. Let's explore the solutions one by one.

Using the Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations. It works for any quadratic equation, regardless of whether it can be easily factored or not. The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a. This formula will provide us with the solutions (the roots) to the quadratic equation. The ± symbol indicates that there are two possible solutions, one where we add the square root term and one where we subtract it. The part inside the square root, b² - 4ac, is called the discriminant. The discriminant tells us about the nature of the roots. The value of the discriminant determines whether the equation has two distinct real roots (if it's positive), one repeated real root (if it's zero), or two complex roots (if it's negative).

For our equation, 2x² - 10x + 13 = 0, we have a = 2, b = -10, and c = 13. Let's plug these values into the quadratic formula:

x = (-(-10) ± √((-10)² - 4 * 2 * 13)) / (2 * 2) x = (10 ± √(100 - 104)) / 4 x = (10 ± √(-4)) / 4

Since we have a negative number under the square root (-4), we know that the equation has complex roots. This means the solutions will involve imaginary numbers.

Understanding Complex Roots

When the discriminant (the value inside the square root) is negative, the quadratic equation has complex roots. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as √(-1). So, let's simplify our solution:

x = (10 ± √(-4)) / 4 x = (10 ± 2i) / 4 x = 10/4 ± 2i/4 x = 5/2 ± 1/2i

Therefore, the two complex roots are x = 5/2 + 1/2i and x = 5/2 - 1/2i. These solutions are complex numbers, which means they are not real numbers and don't appear on the number line. They exist in a two-dimensional plane with real and imaginary axes. The presence of complex roots indicates that the parabola of the quadratic equation doesn't intersect the x-axis.

Completing the Square

Completing the square is another method to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial. This method is particularly useful when factoring isn't straightforward. To complete the square for our equation, 2x² - 10x + 13 = 0, we first divide everything by 2 to make the coefficient of x² equal to 1:

x² - 5x + 13/2 = 0

Next, we move the constant term to the other side: x² - 5x = -13/2. Now, we need to add a value to both sides to complete the square. This value is calculated as (b/2)², where 'b' is the coefficient of the x term. In our case, b = -5, so we add (-5/2)² = 25/4 to both sides:

x² - 5x + 25/4 = -13/2 + 25/4 (x - 5/2)² = -26/4 + 25/4 (x - 5/2)² = -1/4

Taking the square root of both sides:

x - 5/2 = ±√(-1/4) x - 5/2 = ±1/2i

So, x = 5/2 ± 1/2i, which matches the results we got using the quadratic formula. Completing the square is not always the easiest method, especially when dealing with fractions, but it's a valuable technique to know.

Factoring

Factoring is a method where we try to rewrite the quadratic equation as a product of two binomials. If we can factor the quadratic equation, finding the solutions becomes very simple. However, not all quadratic equations can be easily factored. For the equation 2x² - 10x + 13 = 0, it's not easily factorable because the factors don't result in integer values. The factors could involve complex numbers, which aren't always easy to spot at first glance. Factoring is often the quickest method if it's possible, but it doesn't always work. Even though factoring isn't directly applicable here, understanding the concept is useful for simplifying and solving quadratic equations in general.

Conclusion

We've covered a lot of ground! We started by understanding what quadratic equations are, and then we explored three primary methods for solving them: the quadratic formula, completing the square, and factoring. For the equation 13 + 2x² = 10x, we saw that the solutions are complex numbers. This is a great example of how important it is to be comfortable with all the methods and when to use each one. Remember, the best way to become proficient in solving quadratic equations is to practice. Try solving different types of equations using each method. The more you practice, the more comfortable and confident you will become. Keep up the good work, and you'll master quadratic equations in no time. Happy solving, guys!