Solving X² - 8x + 15 = 0 What Is The Value Of X?

by Chloe Fitzgerald 49 views

Hey guys! Let's dive into the fascinating world of quadratic equations and solve a classic problem together. We're going to figure out the value of 'x' in the equation where the square of a number 'x' minus eight times that same number equals -15. Sounds intriguing, right? Don't worry, we'll break it down step by step so it's super easy to follow.

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in our case, 'x') is 2. They generally take the form of ax² + bx + c = 0, where 'a', 'b', and 'c' are constants.

Think of quadratic equations as a mathematical puzzle where we need to find the values of 'x' that make the equation true. These values are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, but we'll be focusing on factoring in this case. Factoring is a powerful technique that allows us to rewrite the quadratic equation as a product of two linear expressions. This makes it easier to identify the roots.

Now, why are quadratic equations so important? Well, they pop up in various real-world scenarios, from physics and engineering to finance and computer science. They can be used to model projectile motion, calculate areas and volumes, and even optimize financial investments. So, understanding how to solve them is a valuable skill to have!

Setting Up the Equation

Okay, let's get back to our specific problem. We're given the equation: x² - 8x = -15. To solve it, we first need to rewrite it in the standard quadratic form, which is ax² + bx + c = 0. To do this, we simply add 15 to both sides of the equation:

x² - 8x + 15 = 0

Great! Now we have our equation in the standard form. We can clearly see that 'a' is 1, 'b' is -8, and 'c' is 15. These coefficients will be important when we factor the equation.

Before we move on, let's take a moment to appreciate the elegance of this equation. It's a simple yet powerful expression that holds the key to finding the hidden values of 'x'. The beauty of mathematics lies in its ability to represent complex relationships in a concise and understandable way. Quadratic equations are a perfect example of this, allowing us to model and solve a wide range of problems with just a few symbols and operations.

Factoring the Quadratic Equation

Now comes the fun part – factoring! Factoring is like reverse multiplication. We need to find two binomials (expressions with two terms) that, when multiplied together, give us our quadratic equation. In other words, we want to find two expressions of the form (x + p) and (x + q) such that:

(x + p)(x + q) = x² - 8x + 15

To find 'p' and 'q', we need to think about two numbers that:

  1. Multiply to give us 'c' (which is 15 in our case).
  2. Add up to give us 'b' (which is -8 in our case).

Let's brainstorm a bit. The factors of 15 are 1 and 15, or 3 and 5. Since we need the numbers to add up to -8, we know they both have to be negative. So, let's try -3 and -5. Do they work?

  • -3 * -5 = 15 (check!)
  • -3 + (-5) = -8 (check!)

Awesome! We found our numbers. So, we can rewrite our quadratic equation as:

(x - 3)(x - 5) = 0

We've successfully factored the quadratic equation! This is a crucial step because it allows us to use the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means that either (x - 3) = 0 or (x - 5) = 0.

Finding the Roots

Now that we've factored the equation, finding the roots is a breeze. We simply set each factor equal to zero and solve for 'x':

  1. x - 3 = 0 Add 3 to both sides: x = 3
  2. x - 5 = 0 Add 5 to both sides: x = 5

And there you have it! We've found the two roots of the quadratic equation. The solutions are x = 3 and x = 5. These are the values of 'x' that make the original equation true. We can verify this by plugging these values back into the original equation and seeing if they satisfy it.

Let's check x = 3:

(3)² - 8(3) + 15 = 9 - 24 + 15 = 0 (check!)

And now let's check x = 5:

(5)² - 8(5) + 15 = 25 - 40 + 15 = 0 (check!)

Both roots work perfectly! We've successfully solved the quadratic equation and found the values of 'x'.

Importance of Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra and has wide-ranging applications in various fields. From physics and engineering to economics and computer science, quadratic equations are used to model and solve a variety of real-world problems.

For example, in physics, quadratic equations are used to describe the motion of projectiles, such as a ball thrown into the air. The height of the ball at any given time can be modeled using a quadratic equation, and solving the equation allows us to determine the time it takes for the ball to reach its maximum height or to hit the ground. In engineering, quadratic equations are used in the design of bridges, buildings, and other structures. The shape of a parabolic arch, which is often used in bridges, is described by a quadratic equation. Solving the equation allows engineers to determine the dimensions of the arch and ensure its stability. In economics, quadratic equations can be used to model supply and demand curves. The equilibrium price and quantity in a market can be found by solving a system of equations, one of which may be quadratic. In computer science, quadratic equations are used in various algorithms, such as those used in computer graphics and image processing.

The ability to solve quadratic equations is also important for understanding more advanced mathematical concepts, such as calculus and differential equations. Many calculus problems involve finding the roots of quadratic equations, and differential equations often involve solving equations that are quadratic in form. Therefore, mastering the techniques for solving quadratic equations is essential for success in higher-level mathematics courses.

Conclusion: The Roots of x² - 8x + 15 = 0

So, to recap, we started with the question: What is the value of x in the quadratic equation where the square of a number x minus eight times that same number is equal to -15? We then rewrote the equation as x² - 8x + 15 = 0, factored it into (x - 3)(x - 5) = 0, and used the zero-product property to find the roots.

Therefore, the solutions to the equation x² - 8x + 15 = 0 are:

  • x = 3
  • x = 5

We did it! We successfully navigated the world of quadratic equations and found the values of 'x' that make the equation true. Remember, practice makes perfect, so keep exploring and solving quadratic equations – you'll become a pro in no time! And hey, if you ever get stuck, don't hesitate to reach out for help. There's a whole community of math enthusiasts out there ready to lend a hand. Keep up the great work, guys!