Solving The Quadratic Equation: 5w^2 = -9w - 4
Hey guys! Let's dive into solving a quadratic equation today. Quadratic equations might seem intimidating at first, but once you grasp the fundamental techniques, you'll be able to tackle them with confidence. In this article, we're going to break down the equation 5w² = -9w - 4 step by step, so you can fully understand the solution. So, grab your pencils and let's get started!
Understanding Quadratic Equations
Before we jump into the solution, let's make sure we're all on the same page. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in this case, w) 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. Recognizing this form is crucial because many solution methods rely on it. These equations pop up all over the place, from physics to engineering, so knowing how to solve them is a seriously valuable skill. You'll see them used to model projectile motion, calculate areas, and even in financial models. The cool thing about quadratic equations is that they can have up to two solutions, which are also known as roots or zeros. These solutions represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis. Finding these solutions helps us understand the behavior and characteristics of the quadratic function. So, with a solid grasp of what quadratic equations are and why they're important, let's jump into tackling our specific problem.
Step 1: Rewrite the Equation in Standard Form
The first crucial step in solving our equation, 5w² = -9w - 4, is to rewrite it in the standard form ax² + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. It’s like organizing your tools before starting a project – it makes the whole process smoother. By getting the equation into standard form, we can easily identify the coefficients a, b, and c, which are essential for applying various solution methods, such as factoring or using the quadratic formula. This standardization not only simplifies the process but also helps prevent errors. So, let’s do it!
To get our equation into standard form, we need to add 9w and 4 to both sides of the equation:
5w² + 9w + 4 = 0
Now we can clearly see that:
- a = 5
- b = 9
- c = 4
With the equation in standard form and the coefficients identified, we're perfectly set up to choose the best method for finding the solutions. Next, we'll explore a powerful technique for solving quadratic equations: factoring.
Step 2: Solving by Factoring
Now that our equation is in the standard form 5w² + 9w + 4 = 0, we can explore solving it by factoring. Factoring is like reverse multiplication – we're trying to find two binomials that, when multiplied together, give us our quadratic equation. It’s a super handy technique when it works, and it can often be quicker than other methods. The idea behind factoring is to break down the quadratic expression into simpler parts, which makes finding the solutions (or roots) much easier. These roots are the values of w that make the equation equal to zero, and they represent the points where the parabola crosses the x-axis. When you get the hang of factoring, you'll start to see patterns and be able to solve many quadratic equations almost by inspection. So, let's dive in and see how we can factor our equation.
To factor the quadratic expression 5w² + 9w + 4, we look for two numbers that multiply to give us the product of a and c (5 * 4 = 20) and add up to b (9). Those numbers are 5 and 4. Now we rewrite the middle term using these numbers:
5w² + 5w + 4w + 4 = 0
Next, we factor by grouping. We group the first two terms and the last two terms:
5w(w + 1) + 4(w + 1) = 0
Notice that (w + 1) is a common factor. We can factor it out:
(5w + 4)(w + 1) = 0
Now we have factored the quadratic equation into two binomials. Next, we'll use the zero-product property to find the solutions.
Step 3: Apply the Zero-Product Property
The zero-product property is a cornerstone for solving equations by factoring. It’s a simple yet powerful concept: If the product of two factors is zero, then at least one of the factors must be zero. In our case, we've factored the quadratic equation into (5w + 4)(w + 1) = 0. This means that either (5w + 4) equals zero, or (w + 1) equals zero, or both. Applying this property allows us to break down the problem into two smaller, easier-to-solve equations. It’s like taking a big problem and splitting it into manageable chunks. This step is crucial because it sets us up to find the individual values of w that satisfy the original equation. So, let’s put this property to work and see what we get.
To find the solutions, we set each factor equal to zero:
- 5w + 4 = 0
- w + 1 = 0
Now we solve each equation separately:
For the first equation, 5w + 4 = 0, we subtract 4 from both sides:
5w = -4
Then, we divide by 5:
w = -4/5
For the second equation, w + 1 = 0, we simply subtract 1 from both sides:
w = -1
So, we have found two solutions for w. Let's summarize our results.
Step 4: State the Solutions
We've reached the final step, guys! After all the hard work, it's time to state the solutions we found. We went through rewriting the equation in standard form, factoring, and applying the zero-product property. Now, we just need to clearly present the values of w that make the original equation true. It’s like putting the finishing touches on a masterpiece – making sure everything is clear and polished. Clearly stating the solutions not only completes the problem but also helps reinforce your understanding of the process. When you see the solutions, you can appreciate how all the steps fit together to give you the answers. So, let’s write down those solutions and celebrate our success!
Our solutions for the equation 5w² = -9w - 4 are:
- w = -4/5
- w = -1
These are the two values of w that satisfy the quadratic equation. If you were to graph the equation, these values would represent the points where the parabola intersects the x-axis.
Alternative Method: The Quadratic Formula
While we successfully solved this equation by factoring, it's worth mentioning another powerful method: the quadratic formula. This formula is like a universal key that unlocks the solutions to any quadratic equation, no matter how complex. It’s especially handy when factoring seems tricky or impossible. The quadratic formula might look intimidating at first, but once you get comfortable with it, it becomes an invaluable tool in your math arsenal. It not only helps you solve equations quickly but also provides a systematic approach that can prevent errors. So, let’s take a peek at what the quadratic formula looks like and how it can be applied.
The quadratic formula is:
w = (-b ± √(b² - 4ac)) / 2a
Where a, b, and c are the coefficients from the standard form of the quadratic equation, ax² + bx + c = 0. To use the quadratic formula for our equation 5w² + 9w + 4 = 0, we plug in the values:
- a = 5
- b = 9
- c = 4
So, the formula becomes:
w = (-9 ± √(9² - 4 * 5 * 4)) / (2 * 5)
Let's simplify it:
w = (-9 ± √(81 - 80)) / 10
w = (-9 ± √1) / 10
w = (-9 ± 1) / 10
This gives us two possible solutions:
- w = (-9 + 1) / 10 = -8 / 10 = -4/5
- w = (-9 - 1) / 10 = -10 / 10 = -1
As you can see, the quadratic formula gives us the same solutions as factoring: w = -4/5 and w = -1.
Conclusion
Awesome job, guys! We've successfully solved the quadratic equation 5w² = -9w - 4 using both factoring and the quadratic formula. We broke down each step, from rewriting the equation in standard form to applying the zero-product property and the quadratic formula. Remember, practice makes perfect, so keep working on these techniques to boost your skills. Quadratic equations might seem tough at first, but with the right approach, you can conquer them all. Whether you prefer factoring or using the quadratic formula, having these tools in your math toolkit will make you a quadratic equation-solving pro. Keep up the great work, and you’ll be acing those math problems in no time!
