Solving Quadratic Equations: Factoring X^2 - 6x = 40

Hey guys! Ever stumbled upon a quadratic equation and felt a little lost? Don't worry, it happens to the best of us. Quadratic equations might seem intimidating at first, but once you understand the basic techniques, they become much easier to handle. In this article, we're going to dive deep into solving the equation x² - 6x = 40 using the factoring method. Factoring is a powerful tool in algebra, and mastering it will not only help you solve equations but also give you a solid foundation for more advanced math topics. So, let’s get started and break down this problem step by step.

What are Quadratic Equations?

Before we jump into the solution, let's quickly recap what quadratic equations are. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x) is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants, and a is not equal to 0. If a were 0, the equation would become linear, not quadratic. Understanding this standard form is crucial because it sets the stage for various methods of solving quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths, and factoring is often the quickest when it's applicable. Now that we've refreshed our understanding of quadratic equations, let's move on to the first step in solving our specific equation.

Step 1: Rearrange the Equation

The first thing we need to do when solving a quadratic equation by factoring (or any method, really) is to set the equation equal to zero. Our original equation is:

x² - 6x = 40

To get it into the standard form, we need to subtract 40 from both sides of the equation. This ensures that one side of the equation is zero, which is essential for the factoring method to work. Think of it like balancing a scale; whatever you do to one side, you must do to the other to keep the equation true. So, let’s subtract 40 from both sides:

x² - 6x - 40 = 0

Now, our equation is in the standard form ax² + bx + c = 0, where a = 1, b = -6, and c = -40. This form allows us to easily identify the coefficients we'll need for factoring. With the equation properly arranged, we can now move on to the heart of the factoring process: finding the right factors.

Step 2: Factoring the Quadratic Expression

Okay, here's where the real fun begins! We need to factor the quadratic expression x² - 6x - 40. Factoring means we're trying to rewrite the expression as a product of two binomials. In other words, we want to find two expressions that, when multiplied together, give us our original quadratic expression. This involves a bit of detective work and pattern recognition.

We're looking for two numbers that:

1. Multiply to the constant term (c), which is -40 in our case.
2. Add up to the coefficient of the x term (b), which is -6.

Think of pairs of factors of -40. We have:

• 1 and -40
• -1 and 40
• 2 and -20
• -2 and 20
• 4 and -10
• -4 and 10
• 5 and -8
• -5 and 8

Now, let's check which of these pairs adds up to -6. Looking at the list, we can see that the pair 4 and -10 fits the bill. 4 multiplied by -10 equals -40, and 4 plus -10 equals -6. Bingo! These are the numbers we need.

Now we can rewrite the quadratic expression using these numbers:

x² - 6x - 40 = (x + 4)(x - 10)

This means that if we were to multiply (x + 4) by (x - 10), we would get back our original expression, x² - 6x - 40. Factoring might seem tricky at first, but with practice, you'll start to recognize these patterns more quickly. Now that we've successfully factored the expression, we're one step closer to solving the equation. Let's see what's next.

Step 3: Apply the Zero Product Property

Alright, we've factored our equation into (x + 4)(x - 10) = 0. Now comes a super important concept called 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. It's a simple but powerful rule that allows us to break down our factored equation into simpler equations.

In our case, we have two factors: (x + 4) and (x - 10). If their product is zero, then either (x + 4) is zero, or (x - 10) is zero, or both. This gives us two separate equations to solve:

1. x + 4 = 0
2. x - 10 = 0

By applying the Zero Product Property, we've transformed one quadratic equation into two linear equations. Linear equations are much easier to solve, as they only involve simple algebraic manipulations. This is where the beauty of factoring shines – it simplifies complex problems into manageable steps. Now, let's solve each of these linear equations to find the values of x that satisfy our original quadratic equation.

Step 4: Solve the Linear Equations

We've got two simple linear equations to solve, which is the home stretch of our problem! Let's tackle them one at a time:

1. x + 4 = 0

   To solve for x, we need to isolate it on one side of the equation. We can do this by subtracting 4 from both sides:

   x + 4 - 4 = 0 - 4

   x = -4

   So, one solution to our original quadratic equation is x = -4.

2. x - 10 = 0

   Similarly, to solve for x in this equation, we add 10 to both sides:

   x - 10 + 10 = 0 + 10

   x = 10

   Therefore, our second solution is x = 10.

And there we have it! We've found the two values of x that make our original equation true. Solving these linear equations is straightforward, and it's a testament to how effectively the factoring method can simplify quadratic equations. Now that we have our solutions, let's make sure they're correct by verifying them.

Step 5: Verify the Solutions

It's always a good idea to check your answers, especially in math. Verifying our solutions ensures that we haven't made any mistakes along the way. To verify, we'll plug each solution back into the original equation x² - 6x = 40 and see if it holds true.

Let's start with x = -4:

(-4)² - 6(-4) = 40

16 + 24 = 40

40 = 40

The equation holds true! So, x = -4 is indeed a solution.

Now, let's check x = 10:

(10)² - 6(10) = 40

100 - 60 = 40

40 = 40

This solution also checks out! Both x = -4 and x = 10 satisfy the original equation, so we can confidently say we've solved it correctly. Verifying solutions is a crucial step in problem-solving, as it provides assurance and reinforces your understanding of the process.

Conclusion

Fantastic job, guys! We've successfully solved the quadratic equation x² - 6x = 40 by factoring. We went through each step in detail:

1. Rearranged the equation to the standard form.
2. Factored the quadratic expression.
3. Applied the Zero Product Property.
4. Solved the resulting linear equations.
5. Verified our solutions.

Factoring is a powerful technique for solving quadratic equations, and mastering it will be incredibly beneficial in your mathematical journey. Remember, practice makes perfect, so try solving more equations using this method. The more you practice, the more comfortable you'll become with recognizing patterns and applying the steps. Keep up the great work, and happy solving!