Solving Quadratic Equations: Finding A+b
Hey math enthusiasts! Let's dive into a classic algebra problem. We're going to tackle the equation x² - x - 90 = 0 and figure out the sum of its solutions. This is a great exercise for solidifying your understanding of quadratic equations, factoring, and the relationship between roots and coefficients. Ready to jump in, guys?
Unveiling the Solutions: Factoring the Quadratic
Alright, first things first: we need to find the solutions (also known as roots) of the quadratic equation x² - x - 90 = 0. There are a couple of ways to do this, but the most straightforward approach here is factoring. Factoring involves breaking down the quadratic expression into two binomials. This is like reverse-engineering a multiplication problem. We're essentially trying to find two numbers that multiply to give us the constant term (-90 in this case) and add up to the coefficient of the x term (-1 in this case).
Think about it: when we expand (x + p)(x + q), we get x² + (p + q)x + pq. So, we need to find 'p' and 'q' such that pq = -90 and p + q = -1. After a bit of mental juggling (or trial and error!), you should realize that the numbers -10 and 9 fit the bill. Because -10 multiplied by 9 equals -90, and -10 plus 9 equals -1.
So, we can rewrite the equation as (x - 10)(x + 9) = 0. Now we can proceed. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, either (x - 10) = 0 or (x + 9) = 0. Solving these simple equations, we get x = 10 and x = -9. These are the two solutions to our quadratic equation.
Let's clarify what's going on here. We start with the given quadratic equation, then we work out the math to find its roots. These are the a and b values the question alludes to. By factoring, we're essentially reversing the process of multiplying two binomial expressions. This helps us isolate the values of x that will make the entire expression equal zero. Essentially, it is where the parabola, or u-shaped curve, crosses the x-axis.
Now, let's designate a = 10 and b = -9. Doesn't matter which order, the final answer will be the same. The essence of the problem lies in understanding that those two values, 10 and -9, are indeed the values that, when plugged into the original equation, make the whole thing equal to zero. This is a fundamental concept in solving quadratic equations. The process of factoring helps us identify these crucial points. When you factor a quadratic equation, you're essentially finding the values of x that make the expression equal zero. It's like finding the secret code to unlock the equation's hidden potential! And the solution to this problem is just around the corner!
Calculating a + b: The Final Step
So, now that we've found our solutions, let's get to the main event: calculating a + b. We've determined that our solutions are 10 and -9. Simple addition gives us the final answer. We just add those values together. The sum of the solutions (a + b) is 10 + (-9) = 1. Congratulations, we solved the problem! That's all there is to it, folks. We tackled the equation, found the roots through factoring, and then computed their sum. Pretty straightforward, right?
This simple problem provides a solid base for understanding more complex algebraic concepts. Knowing how to factor and find the sum and product of roots is important for more complex scenarios. These concepts will be important as you advance through higher level mathematics and sciences. Let's recap what we've learned.
We started with the equation x² - x - 90 = 0. Factoring helped us break it down into (x - 10)(x + 9) = 0. This gave us our roots: x = 10 and x = -9. Then, we simply added these two solutions together to get our final answer. Easy peasy, right?
Exploring Alternative Approaches: The Quadratic Formula
While factoring is a great method for this particular equation, it's not always the easiest or even possible approach. What happens if the quadratic equation doesn't factor neatly? That's where the quadratic formula comes to the rescue! This handy formula can solve any quadratic equation, no matter how complex.
The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a. In this formula, a, b, and c are the coefficients from the standard form of a quadratic equation: ax² + bx + c = 0. For our equation, x² - x - 90 = 0, we have a = 1, b = -1, and c = -90. If you do not remember how to use the quadratic equation, don't worry! You can easily search online how to solve the quadratic equation.
Plugging these values into the formula, we get: x = (1 ± √((-1)² - 4 * 1 * -90)) / (2 * 1). Simplifying this, we get x = (1 ± √(1 + 360)) / 2, or x = (1 ± √361) / 2. Since the square root of 361 is 19, we get x = (1 ± 19) / 2. This gives us two solutions: x = (1 + 19) / 2 = 10 and x = (1 - 19) / 2 = -9, which match our factored results!
This approach shows the versatility of the quadratic formula, and highlights its importance in solving any quadratic equation. The quadratic formula is a universal tool. It works for every quadratic equation. So, whether factoring is easy or not, the quadratic formula always gives the correct answers.
Now, to find a + b, you would simply add the two solutions you get from the quadratic formula. Again, in our case, it's 10 + (-9) = 1. This reinforces the idea that, regardless of how you solve the equation, the sum of the solutions remains constant. The quadratic formula is not just a tool for finding the roots; it also provides a deeper understanding of the relationships within the equation.
The Relationship Between Roots and Coefficients
Here is a handy little shortcut that you might find very useful. You don't always have to solve the quadratic equation to find the sum of the roots! There's a direct relationship between the coefficients of a quadratic equation and the sum and product of its roots.
For a quadratic equation in the form ax² + bx + c = 0, the sum of the roots is given by -b/a, and the product of the roots is given by c/a. For our equation, x² - x - 90 = 0, we have a = 1, b = -1, and c = -90. Therefore, the sum of the roots is -(-1)/1 = 1. And the product of the roots is -90/1 = -90. Isn't that neat?
This method is a real time-saver! It emphasizes the underlying mathematical relationships. This also reinforces the connection between the equation's structure and its solutions. This relationship is not just a formula; it's a fundamental property of quadratic equations. By using this, you can quickly find the sum of the roots without having to solve the equation. It's a great trick to have up your sleeve for any math test.
Practice Makes Perfect: More Examples!
Okay guys, let's try a few more examples to cement our understanding. Here are a couple of practice problems for you to solve on your own. Try them out, and see if you can find the sum of the solutions.
- x² + 5x + 6 = 0
- 2x² - 7x + 3 = 0
For the first equation, you can factor it to (x + 2)(x + 3) = 0. The solutions are -2 and -3. The sum of the solutions is -5. For the second equation, you can factor it to (2x - 1)(x - 3) = 0. The solutions are 1/2 and 3. The sum of the solutions is 7/2.
Feel free to pause here and work through these problems. You can use factoring, the quadratic formula, or the relationship between roots and coefficients. The key is to practice these different methods to familiarize yourself with how to solve these problems.
Conclusion: Mastering Quadratic Equations
We have journeyed through the world of quadratic equations, exploring how to find the solutions to find the sum of the solutions. We learned the importance of factoring, and how to use the quadratic formula. We also discovered a shortcut involving the relationship between roots and coefficients. We saw that no matter how you solve it, you can find the sum of a and b.
Remember, practice is the key to mastering quadratic equations. The more you work through different problems, the more comfortable and confident you'll become. Keep up the great work, and happy solving, folks!