Solving Quadratic Equations: Find 'a' In 5a² - 38a = 16

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Solving Quadratic Equations: Find 'a' in 5a² - 38a = 16

Hey guys! Ever find yourself staring at a quadratic equation and feeling totally lost? Don't worry, we've all been there! Today, we're going to break down how to solve the equation 5a² - 38a = 16. This might look intimidating, but trust me, with a few simple steps, you'll be cracking these problems like a pro. We'll cover everything from rearranging the equation to using the quadratic formula, so buckle up and let's dive in!

Understanding Quadratic Equations

First off, let's get a handle on what we're dealing with. A quadratic equation is basically an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is our variable (in this case, 'a' is our variable, which can be a little confusing, but you'll get it!). The key thing is the x² (or a² in our equation) term – that's what makes it quadratic. Understanding this standard form is super important because it sets us up for using different methods to find the solutions.

Now, why are these equations so important? Well, they pop up all over the place in the real world! Think about physics, engineering, even economics – quadratic equations help us model curves and trajectories, optimize designs, and make predictions. So, mastering them isn't just about acing your math test; it's about building a powerful problem-solving toolset for all sorts of situations. That's why putting in the effort to really understand how to tackle these problems is a fantastic investment in your future skills. Plus, there's something oddly satisfying about finding those solutions and knowing you've conquered a tricky mathematical puzzle!

Setting Up the Equation

Okay, back to our specific problem: 5a² - 38a = 16. The first thing we need to do is get it into that standard form we talked about earlier (ax² + bx + c = 0). This means we need to move everything over to one side of the equation, leaving zero on the other side. How do we do that? Simple! We subtract 16 from both sides. This gives us:

5a² - 38a - 16 = 0

Now we're talking! We've got our equation in the right format. We can clearly see that:

  • a = 5
  • b = -38
  • c = -16

Identifying these coefficients is a crucial step because they're the building blocks for the next phase of our solution. Once you've got these nailed down, you're halfway there! Seriously, taking the time to make sure you've correctly identified a, b, and c will save you a ton of headaches later on. It's like laying a solid foundation for a house – if you get it right, everything else will be much smoother.

Methods to Solve Quadratic Equations

There are a few main ways we can solve quadratic equations:

  1. Factoring: This is like the gold standard if it works! It's often the quickest and most elegant way to find the solutions. But, let's be real, not every quadratic equation is easily factorable. Sometimes, you'll spend ages trying to find the right factors and end up pulling your hair out! So, while it's a great technique to have in your toolbox, you need to be ready to switch gears if factoring isn't playing ball.

  2. Quadratic Formula: This is our trusty backup, the method that always works no matter how messy the equation is. It might look a bit scary at first glance with all those square roots and fractions, but trust me, it's just a matter of plugging in the right numbers and doing the math. Think of it as your mathematical safety net – when all else fails, the quadratic formula is there to save the day! It's especially useful when the coefficients are large or the equation just doesn't seem to want to factor.

  3. Completing the Square: This is a bit of a niche technique, but it's worth knowing about. It's a way of rewriting the quadratic equation into a perfect square trinomial, which then makes it easy to solve. It's not used as often as factoring or the quadratic formula, but it's a powerful method, especially in more advanced mathematical contexts. Plus, understanding completing the square can give you a deeper understanding of how quadratic equations work.

For this particular equation, 5a² - 38a - 16 = 0, factoring might be tricky, so we're going to use the quadratic formula. It's the most reliable method for this problem.

Applying the Quadratic Formula

Alright, let's unleash the quadratic formula! Here it is in all its glory:

a = (-b ± √(b² - 4ac)) / 2a

Don't let it intimidate you! It looks complicated, but it's really just a matter of plugging in our values for 'a', 'b', and 'c' from our equation. Remember:

  • a = 5
  • b = -38
  • c = -16

Now, let's substitute these values into the formula:

a = (-(-38) ± √((-38)² - 4 * 5 * -16)) / (2 * 5)

See? We're just swapping the letters for the numbers. Now, it's time to simplify things, one step at a time. First, let's tackle those negatives and exponents. Remember, a negative times a negative is a positive, and squaring a negative number also gives you a positive result. So, we get:

a = (38 ± √(1444 + 320)) / 10

Next up, let's add the numbers under the square root:

a = (38 ± √1764) / 10

Now we need to find the square root of 1764. If you've got a calculator handy, this is a piece of cake! If not, you might need to do a bit of prime factorization or estimation. But the good news is, the square root of 1764 is a whole number:

√1764 = 42

So our equation becomes:

a = (38 ± 42) / 10

We're in the home stretch now! Notice the ± sign? This means we actually have two solutions, one where we add 42 and one where we subtract 42. Let's calculate them separately.

Finding the Solutions

Okay, let's break it down and find our two solutions for 'a'.

Solution 1: Using the Plus Sign

First, we'll use the plus sign in the ± symbol:

a = (38 + 42) / 10

This simplifies to:

a = 80 / 10

And finally:

a = 8

So, our first solution is a = 8. That's one down, one to go!

Solution 2: Using the Minus Sign

Now, let's use the minus sign:

a = (38 - 42) / 10

This gives us:

a = -4 / 10

Which we can simplify to:

a = -2/5

So, our second solution is a = -2/5. And there we have it! We've found both possible values for 'a'.

The Final Answer

We've successfully navigated the quadratic equation 5a² - 38a = 16 and found our solutions! To recap, we rearranged the equation into standard form, identified our coefficients, used the quadratic formula, and carefully simplified to arrive at our answers. The solutions are:

  • a = 8
  • a = -2/5

These are the two values of 'a' that make the original equation true. You can always check your work by plugging these values back into the original equation and seeing if both sides balance out. It's a great way to make sure you haven't made any silly mistakes along the way!

Solving quadratic equations might seem daunting at first, but with practice and a solid understanding of the steps, you can conquer any quadratic equation that comes your way. Remember, the key is to break down the problem into smaller, manageable steps, and don't be afraid to use the tools at your disposal, like the quadratic formula. Keep practicing, and you'll become a quadratic equation-solving ninja in no time!