Solving G(x) = 6x^2 + 23x - 4 Finding When G(x) = 0

Hey guys! Let's dive into solving a quadratic equation. We've got this function, g(x) = 6x² + 23x - 4, and the challenge is to figure out when g(x) equals zero. In other words, we need to find the values of x that make this equation true. This is a classic problem in algebra, and there are a few ways we can tackle it. We'll primarily focus on factoring and using the quadratic formula, both powerful tools for cracking these types of problems.

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (x in our case) 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. Our equation, g(x) = 6x² + 23x - 4, perfectly fits this form, with a = 6, b = 23, and c = -4. Solving a quadratic equation means finding its roots or solutions, which are the values of x that satisfy the equation. These roots are also the x-intercepts of the parabola represented by the quadratic function.

Solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications in various fields. From physics and engineering to economics and computer science, quadratic equations pop up everywhere. For example, they can be used to model the trajectory of a projectile, the shape of a suspension bridge, or the optimization of business processes. Mastering the techniques for solving quadratic equations is therefore crucial for anyone pursuing studies or a career in these areas. We'll explore two primary methods: factoring and the quadratic formula. Factoring involves breaking down the quadratic expression into a product of two linear expressions, while the quadratic formula provides a direct solution for x in terms of the coefficients a, b, and c. Each method has its advantages and disadvantages, and the choice of which method to use often depends on the specific equation at hand. By understanding both methods, we'll be well-equipped to solve a wide variety of quadratic equations efficiently and accurately.

Method 1 Factoring

One way to solve 6x² + 23x - 4 = 0 is by factoring. Factoring involves breaking down the quadratic expression into a product of two binomials. If we can find two binomials that multiply together to give us 6x² + 23x - 4, we can then set each binomial equal to zero and solve for x. This works because if the product of two factors is zero, then at least one of the factors must be zero. So, let's try to factor this expression.

The key to factoring this quadratic is to find two numbers that multiply to a c (which is 6 * -4 = -24) and add up to b (which is 23). Think of it like a puzzle – we need to find the right combination of numbers that fit these conditions. After a bit of thought, we can see that the numbers 24 and -1 work perfectly. 24 multiplied by -1 equals -24, and 24 plus -1 equals 23. Now, we'll rewrite the middle term (23x) using these two numbers. We can rewrite the equation as 6x² + 24x - x - 4 = 0. Notice how we've split the 23x term into 24x and -x, using the numbers we just found. This step is crucial because it allows us to group the terms and factor by grouping.

Next, we'll factor by grouping. We'll group the first two terms and the last two terms together: (6x² + 24x) + (-x - 4) = 0. Now, we'll factor out the greatest common factor (GCF) from each group. From the first group, 6x² + 24x, the GCF is 6x. Factoring out 6x, we get 6x(x + 4). From the second group, -x - 4, the GCF is -1. Factoring out -1, we get -1(x + 4). So, our equation now looks like 6x(x + 4) - 1(x + 4) = 0. Notice that both terms now have a common factor of (x + 4). We can factor out (x + 4) from the entire equation, giving us (x + 4)(6x - 1) = 0. This is the factored form of our original quadratic expression. Now we're in the home stretch!

Applying the Zero Product Property

Now that we have factored the quadratic equation into (x + 4)(6x - 1) = 0, we can use the zero product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if ab = 0, then either a = 0 or b = 0 (or both). Applying this to our factored equation, we have two possibilities: either (x + 4) = 0 or (6x - 1) = 0. Let's solve each of these equations separately.

First, let's solve x + 4 = 0. To isolate x, we subtract 4 from both sides of the equation. This gives us x = -4. So, one solution to our quadratic equation is x = -4. Now, let's solve the second equation, 6x - 1 = 0. To isolate x, we first add 1 to both sides of the equation, giving us 6x = 1. Then, we divide both sides by 6 to get x = 1/6. So, our second solution is x = 1/6. Therefore, the solutions to the quadratic equation 6x² + 23x - 4 = 0 are x = -4 and x = 1/6. We've successfully found the values of x that make g(x) = 0 by using the factoring method. Factoring is a powerful technique for solving quadratic equations, but it's not always the easiest or most straightforward method, especially when the coefficients are large or the factors are not immediately obvious. That's where the quadratic formula comes in handy, which we'll explore next.

Method 2 The Quadratic Formula

Another powerful way to solve quadratic equations is by using the quadratic formula. This formula provides a direct solution for x in any quadratic equation of the form ax² + bx + c = 0. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

It looks a bit intimidating, but it's a trusty tool that always works, no matter how messy the equation is! Let's apply this formula to our equation, g(x) = 6x² + 23x - 4 = 0. Remember, in this case, a = 6, b = 23, and c = -4. Now, we'll carefully plug these values into the quadratic formula.

First, let's calculate the discriminant, which is the part under the square root: b² - 4ac. Plugging in our values, we get 23² - 4 * 6 * -4. This simplifies to 529 + 96, which equals 625. The discriminant is a crucial part of the quadratic formula because it tells us about the nature of the roots. If the discriminant is positive, like in our case, the equation has two distinct real roots. If it's zero, the equation has one real root (a repeated root), and if it's negative, the equation has two complex roots. Now that we know the discriminant is 625, we can take its square root, which is 25. This will make our calculations easier.

Now, let's plug the values into the entire quadratic formula: x = (-23 ± √625) / (2 * 6). This simplifies to x = (-23 ± 25) / 12. Remember the ± sign means we have two possible solutions: one with the plus sign and one with the minus sign. Let's calculate the first solution using the plus sign: x = (-23 + 25) / 12. This simplifies to x = 2 / 12, which further simplifies to x = 1/6. This is one of the solutions we found earlier using factoring! Now, let's calculate the second solution using the minus sign: x = (-23 - 25) / 12. This simplifies to x = -48 / 12, which equals x = -4. This is also the other solution we found using factoring. So, using the quadratic formula, we've confirmed that the solutions to the equation 6x² + 23x - 4 = 0 are x = 1/6 and x = -4. The quadratic formula is a powerful tool because it works for any quadratic equation, even those that are difficult or impossible to factor. It's a reliable method to have in your mathematical toolkit.

Comparing the Methods

We've now solved the equation g(x) = 6x² + 23x - 4 = 0 using both factoring and the quadratic formula. Let's take a moment to compare these two methods and see when one might be preferred over the other. Factoring, as we saw, involves breaking down the quadratic expression into a product of two binomials. This method is often quicker and more straightforward when the quadratic expression can be easily factored. It relies on our ability to recognize patterns and find the right combination of numbers that multiply to ac and add up to b. However, not all quadratic equations are easily factorable. When the coefficients are large, or the roots are not rational numbers, factoring can become quite challenging and time-consuming. In these cases, the quadratic formula provides a more reliable and efficient approach.

The quadratic formula, on the other hand, is a general formula that works for any quadratic equation, regardless of whether it's easily factorable or not. It involves plugging the coefficients a, b, and c into a specific formula and performing the calculations. While the formula itself might seem a bit intimidating at first, it's a straightforward process once you get the hang of it. The quadratic formula is particularly useful when dealing with quadratic equations that have irrational or complex roots, as these are difficult to find through factoring. However, the quadratic formula can sometimes involve more calculations than factoring, especially if the coefficients are simple and the equation is easily factorable. So, the choice between factoring and the quadratic formula often depends on the specific equation at hand. If the equation looks easily factorable, that might be the quickest route. But if factoring seems difficult, or if you're unsure, the quadratic formula is always a reliable option.

Conclusion

So, there you have it, guys! We've successfully solved the quadratic equation g(x) = 6x² + 23x - 4 = 0 using two different methods: factoring and the quadratic formula. We found that the solutions are x = -4 and x = 1/6. Both methods are valuable tools for solving quadratic equations, and understanding when to use each one can make your problem-solving process much more efficient. Remember, factoring is great for equations that are easily factorable, while the quadratic formula is a reliable workhorse that works for any quadratic equation. Keep practicing with these methods, and you'll become a quadratic equation-solving pro in no time!