Solving Quadratic Equations Finding Roots Of X^2 + 3x - 3 = 0
Hey guys! Let's dive into a fun math problem today. We're going to find the roots of the quadratic equation x² + 3x - 3 = 0. This might sound intimidating, but don't worry, we'll break it down step by step. We have a few options presented to us, and our mission is to figure out which two of these values for x actually make the equation true.
Understanding Quadratic Equations and Their Roots
Before we jump into solving, let's quickly recap what quadratic equations and their roots are all about. In the realm of mathematics, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, x) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The equation we're dealing with, x² + 3x - 3 = 0, perfectly fits this form, with a = 1, b = 3, and c = -3. The roots of a quadratic equation are the values of x that satisfy the equation, meaning when you plug these values into the equation, the left side equals the right side (which is zero in this case). These roots are also the x-intercepts of the parabola represented by the quadratic equation when graphed. A quadratic equation can have two distinct real roots, one real root (which is a repeated root), or two complex roots.
To find these roots, we can use several methods, including factoring, completing the square, or the quadratic formula. Factoring is a straightforward method when the quadratic expression can be easily factored, but it's not always applicable. Completing the square is a method that transforms the equation into a perfect square trinomial, making it easier to solve. The quadratic formula is a universal method that works for any quadratic equation, regardless of whether it can be factored or not. Given its versatility, we'll use the quadratic formula to solve our equation. The quadratic formula is a powerful tool, guys, and it states that for an equation ax² + bx + c = 0, the roots are given by: x = (-b ± √(b² - 4ac)) / (2a). This formula might look a bit scary, but it's just a matter of plugging in the correct values and simplifying. Remember this formula, it is very important for math.
Applying the Quadratic Formula to Our Equation
Now, let's put the quadratic formula to work on our equation, x² + 3x - 3 = 0. As we identified earlier, a = 1, b = 3, and c = -3. Plugging these values into the quadratic formula, we get: x = (-3 ± √(3² - 4 * 1 * -3)) / (2 * 1). The next step is to simplify the expression. First, let's deal with the part under the square root: 3² - 4 * 1 * -3 = 9 + 12 = 21. So, our equation now looks like this: x = (-3 ± √21) / 2. This gives us two possible solutions for x: x = (-3 + √21) / 2 and x = (-3 - √21) / 2. These are the two roots of our quadratic equation. Guys, you see how simple it can be if we know the formula.
Matching the Solutions with the Given Options
Okay, we've found the roots using the quadratic formula. Now, let's compare our solutions with the options provided. We have: A. x = (-3 + √21) / 2 B. x = (-3 - √21) / 2 C. x = (-3 + √3) / 2 D. x = (-3 - √3) / 2. Looking at our calculated roots, we can see that options A and B perfectly match our solutions. Option A, x = (-3 + √21) / 2, is one of the roots we found, and option B, x = (-3 - √21) / 2, is the other root. Options C and D have √3 instead of √21, so they are not the correct solutions. Therefore, the two values of x that are roots of the equation x² + 3x - 3 = 0 are A and B. So, the correct answers are A and B. We have successfully decoded the roots of the equation. Remember, the key is to understand the quadratic formula and apply it carefully.
Why Other Options Are Incorrect
To solidify our understanding, let's briefly discuss why options C and D are incorrect. These options, x = (-3 + √3) / 2 and x = (-3 - √3) / 2, look similar to our correct solutions, but the crucial difference lies in the value under the square root. We calculated √21 using the quadratic formula, specifically the discriminant (b² - 4ac). If we were to plug in √3 into our original equation, it wouldn't satisfy the equation x² + 3x - 3 = 0. This is because the discriminant determines the nature of the roots, and a different discriminant will lead to different roots. Guys, it's like using the wrong key for a lock; it might look similar, but it won't open the door.
Another way to think about it is to visualize the graph of the quadratic equation. The roots are the points where the parabola intersects the x-axis. Changing the value under the square root effectively changes the shape and position of the parabola, thus changing its x-intercepts. Options C and D would correspond to a different parabola altogether. This highlights the importance of accurately applying the quadratic formula and paying close attention to the values involved. Math is all about precision, and even a small change can lead to a different outcome. So, always double-check your calculations and make sure you're using the correct values in the formula. We know that options C and D are not the correct solutions because they do not satisfy the original equation. Math is fun, isn't it?
Tips and Tricks for Solving Quadratic Equations
Solving quadratic equations can become second nature with practice. Here are a few tips and tricks to help you master this skill: 1. Memorize the Quadratic Formula: This is your go-to tool for solving any quadratic equation. Write it down, practice using it, and make it your friend. 2. Identify a, b, and c Correctly: Pay close attention to the coefficients of the quadratic equation. Make sure you correctly identify a, b, and c, including their signs. A common mistake is to misidentify the sign of c, so be extra careful. 3. Simplify Carefully: Take your time when simplifying the expression under the square root and the entire formula. Avoid making careless errors, as even a small mistake can lead to an incorrect answer. 4. Check Your Solutions: Once you've found the roots, plug them back into the original equation to verify that they satisfy the equation. This is a great way to catch any errors you might have made along the way. 5. Practice, Practice, Practice: The more you practice solving quadratic equations, the more comfortable and confident you'll become. Work through various examples, and don't be afraid to ask for help if you get stuck. 6. Consider Factoring First: Before jumping to the quadratic formula, see if the equation can be easily factored. Factoring can be a quicker method when applicable. 7. Understand the Discriminant: The discriminant (b² - 4ac) tells you about the nature of the roots. If it's positive, there are two distinct real roots; if it's zero, there's one real root (a repeated root); and if it's negative, there are two complex roots. Guys, using these tricks will make you a pro in solving these equations.
Conclusion
In conclusion, we successfully found the two roots of the quadratic equation x² + 3x - 3 = 0 using the quadratic formula. We identified the correct options as A. x = (-3 + √21) / 2 and B. x = (-3 - √21) / 2. We also discussed why the other options were incorrect and shared some tips and tricks for solving quadratic equations. Remember, guys, math can be challenging, but with a solid understanding of the concepts and consistent practice, you can conquer any problem. Keep up the great work, and happy problem-solving! If you have any more questions or want to explore other math topics, feel free to ask. Let's keep learning and growing together. You've got this!
