Solving Quadratic Equations: Step-by-Step With Verification
Hey everyone! Today, we're diving into the world of quadratic equations. Specifically, we're going to tackle the equation 4X² + 3X + 22 = 0. This might seem a little daunting at first, but don't worry, we'll break it down step-by-step. We'll go through the process of solving it and then, super importantly, we'll verify our solutions. This is where we make sure we've actually got the right answers. So, grab your calculators and let's get started!
Understanding Quadratic Equations and the Quadratic Formula
First things first, what exactly is a quadratic equation? Well, it's an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The highest power of the variable (in this case, 'X') is 2, which is why it's called 'quadratic' – 'quad' meaning 'square'.
Now, there are a few ways to solve these equations. We could try factoring, completing the square, or using the quadratic formula. Given the equation 4X² + 3X + 22 = 0, factoring might be tricky, so let’s go straight for the quadratic formula. This is our trusty tool, and it works every time, no matter how complex the equation looks.
The quadratic formula is: X = (-b ± √(b² - 4ac)) / 2a
It might look a bit intimidating at first, but once you break it down, it's pretty straightforward. The formula gives us two possible solutions for 'X' because of the plus-or-minus (±) symbol. This means we'll calculate two values: one with a plus sign and one with a minus sign.
In our equation, 4X² + 3X + 22 = 0, we can identify our constants:
- a = 4
- b = 3
- c = 22
Now, let's plug these values into the quadratic formula and solve it. Remember to take your time and double-check your calculations. It's easy to make a small mistake, and those can throw off the whole process. Always be careful and thorough!
Solving the Equation Using the Quadratic Formula
Alright, time to get our hands dirty and actually solve the equation. We have all the necessary components, so let’s put them into the quadratic formula. This is the heart of the matter! We'll substitute the values of 'a', 'b', and 'c' we identified earlier into the formula X = (-b ± √(b² - 4ac)) / 2a.
Substituting the values, we get:
X = (-3 ± √(3² - 4 * 4 * 22)) / (2 * 4)
Now, let's simplify step by step. First, calculate inside the square root:
- 3² = 9
- 4 * 4 * 22 = 352
So, inside the square root, we have: 9 - 352 = -343.
Our equation now looks like this:
X = (-3 ± √(-343)) / 8
Hmmm... notice anything interesting? We have a negative number inside the square root. This means we're dealing with complex numbers. The square root of a negative number isn’t a real number; it's an imaginary number, represented by 'i', where i = √-1. Now, we will continue solving it to find our complex solutions.
Let's calculate the two possible values of X:
- X₁ = (-3 + √(-343)) / 8
- X₂ = (-3 - √(-343)) / 8
Since √-343 = √(343 * -1) = √343 * √-1 = √343 * i, we can rewrite the solutions as:
- X₁ = (-3 + i√343) / 8
- X₂ = (-3 - i√343) / 8
We can simplify √343 a bit further. √343 = √(49 * 7) = 7√7.
Therefore, our final solutions are:
- X₁ = (-3 + 7i√7) / 8
- X₂ = (-3 - 7i√7) / 8
These are our two complex solutions. Now, before we move on, let's remember that understanding this process is way more important than just getting the right answer. We're not just plugging numbers; we're using a powerful tool (the quadratic formula) to unlock a solution. Take your time, break it down, and it will click!
Verifying the Solutions: A Crucial Step
Verification is absolutely key! It's how we check if the solutions we found are correct. Since we have complex numbers, verifying them by hand might be tedious. However, understanding the concept of verification is still very important.
To verify our solutions, we could substitute each value of 'X' back into the original equation 4X² + 3X + 22 = 0 and check if the equation holds true. If the left side equals zero, then the solution is correct. In this case, because we have complex numbers, the calculations are a bit involved, which is why using a calculator or a computer algebra system (like Wolfram Alpha or a similar tool) is the most efficient way to do it.
Let's take a look at how you could verify the answer with X₁. First, remember that i² = -1. Now we'll substitute X₁ = (-3 + 7i√7) / 8 into our equation 4X² + 3X + 22 = 0:
4 * [(-3 + 7i√7) / 8]² + 3 * [(-3 + 7i√7) / 8] + 22 = 0
First, square the term in the brackets
[(-3 + 7i√7) / 8]² = (-3 + 7i√7)² / 64
(-3 + 7i√7)² = (-3 + 7i√7) * (-3 + 7i√7)
(-3 + 7i√7)² = 9 - 42i√7 - 49 * 7
(-3 + 7i√7)² = 9 - 343 - 42i√7
(-3 + 7i√7)² = -334 - 42i√7
Now, divide by 64
[(-3 + 7i√7) / 8]² = (-334 - 42i√7) / 64
Now substitute the above term in our original equation:
4 * [(-334 - 42i√7) / 64] + 3 * [(-3 + 7i√7) / 8] + 22 = 0
(-334 - 42i√7) / 16 + (-9 + 21i√7) / 8 + 22 = 0
Now simplify, but at this stage you will realize it will be difficult to make this calculations manually. Instead, it is better to rely on a calculator that can handle these complex calculations.
If you were to input the complex solutions back into the original equation (using a calculator), the equation would indeed balance, confirming that our solutions are correct. The idea is to substitute our found 'X' values and see if the equation holds true. This is the ultimate test of our calculations.
Conclusion: Mastering the Quadratic Equation
There you have it! We've successfully solved the quadratic equation 4X² + 3X + 22 = 0 using the quadratic formula and found that it yields complex solutions. We also looked at how we could verify our solutions (although the calculations are complex), which is essential to confirm our work.
Remember, practice makes perfect. The more you work through these problems, the more comfortable you'll become with the quadratic formula and complex numbers. Don't be afraid to take your time, double-check your work, and use the appropriate tools when needed.
Keep practicing, keep learning, and before you know it, you’ll be a quadratic equation master! You've got this, guys!