Solving Quadratic Equations: Find X For 14 + 2x^2 = 24

Hey guys! Today, let's dive into solving a quadratic equation. Quadratic equations might sound intimidating, but trust me, they're totally manageable once you break them down step by step. We're going to tackle the equation 14 + 2x^2 = 24 and find all the possible values of x that make this equation true. We'll simplify it along the way to get the answers in their simplest form. So, grab your thinking caps, and let's get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's quickly recap what quadratic equations are all about. In essence, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, x) is 2. The standard form of a quadratic equation is:

ax^2 + bx + c = 0

Where:

• a, b, and c are constants (numbers), and a is not equal to 0.
• x is the variable we're trying to solve for.

Quadratic equations pop up everywhere in math and real-world applications, from physics to engineering to finance. Knowing how to solve them is a super valuable skill. There are several methods to solve quadratic equations, and we'll use one of the most straightforward methods for our equation today: isolating the variable and using the square root property. This method works great when we can easily isolate the x² term.

Why Solve Quadratic Equations?

Solving quadratic equations isn't just an abstract math exercise; it has real-world implications. Think about situations where you need to model parabolic paths, such as the trajectory of a ball thrown in the air, or design structures involving curves. Quadratic equations are your go-to tool. In engineering, they help in designing bridges and arches, ensuring stability and optimal performance. In physics, they’re crucial for calculating projectile motion and understanding energy conservation. Even in finance, quadratic equations can be used to model growth and decay scenarios, like calculating the break-even point for a business or determining investment returns. Understanding how to solve these equations empowers you to tackle a variety of practical problems.

Key Concepts in Quadratic Equations

Before diving into our example, it's beneficial to understand a few key concepts related to quadratic equations. First, remember the standard form: ax^2 + bx + c = 0. This form is not just a formality; it helps in identifying the coefficients a, b, and c, which are crucial for various solving methods like the quadratic formula. The discriminant, given by the formula b^2 - 4ac, is another critical concept. It tells us about the nature of the roots (solutions) of the equation. If the discriminant is positive, there are two distinct real roots; if it’s zero, there is exactly one real root; and if it’s negative, there are no real roots. This concept is vital because it guides our expectations about the solution. For our equation, 14 + 2x^2 = 24, we'll focus on isolating the x^2 term, but understanding these underlying principles gives you a more comprehensive grasp of quadratic equations in general. Mastery of these concepts makes tackling more complex problems much smoother.

Step-by-Step Solution for 14 + 2x^2 = 24

Okay, let's get to the main event: solving the equation 14 + 2x^2 = 24. We'll break it down into simple steps so it's super clear.

Step 1: Isolate the Term with x^2

Our goal here is to get the term with x² by itself on one side of the equation. To do this, we need to get rid of the 14 that's being added on the left side. We can do this by subtracting 14 from both sides of the equation. Remember, what we do to one side, we have to do to the other to keep things balanced!

14 + 2x^2 - 14 = 24 - 14

This simplifies to:

2x^2 = 10

Step 2: Divide to Isolate x^2

Now we have 2x^2 = 10. We want to get x² all by itself, so we need to get rid of the 2 that's multiplying it. We can do this by dividing both sides of the equation by 2:

(2x^2) / 2 = 10 / 2

This simplifies to:

x^2 = 5

Step 3: Take the Square Root of Both Sides

We're almost there! We have x² = 5, but we want to find x, not x². To do this, we need to take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots because both positive and negative numbers, when squared, will give a positive result.

√ (x^2) = ± √5

This gives us:

x = ± √5

Step 4: Simplify the Solution (if needed)

In this case, √5 is already in its simplest form because 5 is a prime number (it's only divisible by 1 and itself). So, we don't need to simplify any further.

Step 5: State the Solutions

So, our solutions are:

x = √5 and x = -√5

These are the two values of x that make the original equation 14 + 2x^2 = 24 true.

Alternative Methods for Solving Quadratic Equations

While we used the isolation and square root method in this example, it's good to know there are other tools in your quadratic equation-solving toolbox. Two common methods include factoring and using the quadratic formula. Let's briefly touch on these.

Factoring

Factoring involves expressing the quadratic equation in the form (px + q)(rx + s) = 0, where p, q, r, and s are constants. This method is particularly useful when the quadratic equation can be easily factored. For instance, the equation x^2 + 5x + 6 = 0 can be factored into (x + 2)(x + 3) = 0, leading to solutions x = -2 and x = -3. Factoring can be quicker than other methods when it works, but it’s not always straightforward, especially if the equation has irrational or complex roots. Recognizing when an equation is easily factorable is a valuable skill that can save time and effort.

The Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of its complexity. Given a quadratic equation in the standard form ax^2 + bx + c = 0, the quadratic formula is:

x = [-b ± √(b^2 - 4ac)] / (2a)

This formula might look intimidating at first, but it's a powerful tool. You simply plug in the coefficients a, b, and c from your equation, and it spits out the solutions. The quadratic formula is especially useful when the equation is not easily factorable or when you need a precise solution. The discriminant, b^2 - 4ac, within the formula also tells you about the nature of the roots, as mentioned earlier. Learning and practicing the quadratic formula ensures you can tackle any quadratic equation that comes your way.

Real-World Applications

Quadratic equations aren't just confined to textbooks; they have numerous real-world applications that make understanding them essential. In physics, they're used to model projectile motion. Imagine you're calculating how far a ball will travel when thrown at a certain angle and speed; you'll likely be using a quadratic equation. The parabolic path that the ball follows is described by a quadratic function. Engineers use quadratic equations to design bridges and arches, ensuring structural stability and efficient load distribution. The curves and shapes often seen in architecture are frequently based on quadratic functions.

In the business world, quadratic equations can help in analyzing profit and cost curves. For instance, businesses use them to find the break-even point where revenue equals expenses. Financial analysts might use quadratic models to predict investment returns or analyze market trends. Even in computer graphics, quadratic equations are used to create curves and surfaces in 3D modeling and animation. These diverse applications highlight the importance of mastering quadratic equations for anyone pursuing careers in science, technology, engineering, mathematics, and even business and arts.

Conclusion

And that's it! We've successfully solved the quadratic equation 14 + 2x^2 = 24 and found that the values of x are √5 and -√5. Remember, the key is to isolate the x² term, take the square root of both sides, and don't forget to consider both positive and negative roots. Guys, I hope this step-by-step guide helped you understand how to tackle quadratic equations. Keep practicing, and you'll become a pro in no time! Whether you’re aiming for top grades in math or tackling real-world problems, understanding quadratic equations is a valuable asset. So, keep those problem-solving skills sharp, and you'll be well-equipped to face any mathematical challenge.