Solving Quadratics: Completing The Square With -3g^2+12g=-57

Hey guys! Today, we're diving deep into a crucial technique in algebra: completing the square. Specifically, we're going to tackle the quadratic equation $-3g^2 + 12g = -57$ using this method. Trust me, mastering completing the square will seriously level up your equation-solving skills. So, let’s break it down step by step and make sure you've got it down pat. Let’s get started!

Understanding Completing the Square

Before we jump into the specific equation, let's quickly recap what completing the square actually means. In essence, it's a technique used to rewrite a quadratic equation in the form $ax^2 + bx + c = 0$ into the vertex form $a(x - h)^2 + k = 0$. This vertex form is super handy because it directly reveals the vertex (h, k) of the parabola represented by the quadratic equation. But more importantly for our current mission, it allows us to easily solve for the variable, in our case, g. Why is this method so powerful? Because it transforms a potentially messy quadratic into a perfect square trinomial, which we can then factor and solve much more easily. Think of it as turning a complex puzzle into something manageable.

Completing the square is especially useful when the quadratic equation doesn't factor easily or when we need to find the vertex of the parabola. It’s a versatile tool that comes in clutch in many algebraic situations. The beauty of completing the square lies in its systematic approach. By following a set of steps, we can manipulate any quadratic equation into a solvable form. This method is not just about finding the solution; it’s about understanding the structure of quadratic equations and how to manipulate them effectively. Understanding this process also lays a strong foundation for more advanced mathematical concepts, so pay close attention!

Furthermore, when dealing with quadratic equations, especially in higher mathematics, recognizing the appropriate method is crucial. Completing the square provides a robust alternative when factoring is cumbersome or when the quadratic formula seems like overkill. It enhances your problem-solving toolkit and offers a deeper insight into quadratic behavior. So, stick with me as we unravel the mystery behind completing the square and make it a reliable technique in your mathematical arsenal. Remember, practice makes perfect, and the more you use this method, the more natural it will become.

Step-by-Step Solution for $-3g^2 + 12g = -57$

Now, let's apply completing the square to our equation: $-3g^2 + 12g = -57$. We’ll go through each step meticulously so you can follow along and understand the process.

Step 1: Make the Leading Coefficient 1

The golden rule of completing the square is that the coefficient of the $g^2$ term must be 1. In our equation, we have $-3g^2$, so we need to divide the entire equation by -3. This gives us:

$g^2 - 4g = 19$

Dividing each term ensures we maintain the equality while simplifying the quadratic term. This step is crucial because it sets the stage for creating a perfect square trinomial. A leading coefficient of 1 makes the subsequent steps of finding the constant term to complete the square straightforward. It's like laying the foundation for a sturdy building; if this step isn't done correctly, the rest of the process becomes much harder. This simple division transforms our equation into a more manageable form, ready for the next steps. Always double-check your division to avoid errors that could propagate through the rest of the solution. Trust me, it’s worth the extra few seconds to ensure accuracy here.

Step 2: Complete the Square

Here’s where the magic happens! We need to add a value to both sides of the equation to create a perfect square trinomial on the left side. To find this value, we take half of the coefficient of our g term (which is -4), square it, and add it to both sides. Half of -4 is -2, and (-2)^2 is 4. So, we add 4 to both sides:

$g^2 - 4g + 4 = 19 + 4$

Adding the same value to both sides ensures that the equation remains balanced. The number we're adding, 4, is precisely chosen to create a perfect square trinomial, which can be factored into a binomial squared. This is the heart of the completing the square method. By adding this carefully calculated value, we are essentially reshaping the left side of the equation into a form that we can easily manipulate. Think of it as finding the missing piece of a puzzle that transforms the equation into a more recognizable and solvable form. It's a smart move that simplifies the problem considerably.

Step 3: Factor and Simplify

Now, the left side of the equation is a perfect square trinomial. We can factor it as follows:

$(g - 2)^2 = 23$

The expression $g^2 - 4g + 4$ perfectly factors into $(g - 2)^2$. This is the key outcome of completing the square. We've transformed a complex quadratic expression into a simple, squared binomial. The right side of the equation, $19 + 4$, simplifies to 23. So, we now have a much more manageable equation: $(g - 2)^2 = 23$. This step highlights the elegance of completing the square. By adding the correct constant, we’ve converted the quadratic into a form that’s easy to solve. Factoring the perfect square trinomial is a critical step, as it sets up the equation for the final isolation of the variable. This is where all our hard work starts to pay off. The simplicity of $(g - 2)^2$ makes the remaining steps straightforward and less prone to errors.

Step 4: Take the Square Root

To get rid of the square, we take the square root of both sides of the equation. Remember to consider both the positive and negative square roots:

$g - 2 = \pm\sqrt{23}$

Taking the square root introduces both positive and negative solutions because both $(\sqrt{23})^2$ and $(-\sqrt{23})^2$ equal 23. It's a crucial step that ensures we don't miss any possible solutions. The $\pm$ symbol is a shorthand way of representing both the positive and negative roots, saving us from writing two separate equations. By doing this, we’re one step closer to isolating g and finding our final answers. This step is where the quadratic nature of the equation truly comes into play, as quadratic equations typically have two solutions. Always remember to include both the positive and negative roots to ensure you've captured all possible answers.

Step 5: Solve for g

Finally, to isolate g, we add 2 to both sides of the equation:

$g = 2 \pm \sqrt{23}$

Adding 2 to both sides isolates g, giving us our solutions. The expression $2 \pm \sqrt{23}$ represents two distinct solutions: $2 + \sqrt{23}$ and $2 - \sqrt{23}$. These are the roots of the original quadratic equation. We've successfully solved for g by completing the square! This final step is the culmination of all our previous efforts. By carefully manipulating the equation, we've arrived at a clear and concise solution. The $2 \pm \sqrt{23}$ form is the exact solution, and it can be approximated as decimal values if needed. This is the moment where we can pat ourselves on the back for successfully navigating the problem.

The Final Solutions

So, the solutions to the equation $-3g^2 + 12g = -57$ are:

$g = 2 + \sqrt{23}$ and $g = 2 - \sqrt{23}$

These are the exact solutions. If we need approximate decimal values, we can use a calculator:

$g \approx 6.796$ and $g \approx -2.796$

These decimal approximations give us a more tangible sense of the values of g that satisfy the equation. However, it's important to recognize that the exact solutions, $2 + \sqrt{23}$ and $2 - \sqrt{23}$, provide the most accurate representation of the roots. Understanding both the exact and approximate forms is valuable, depending on the context of the problem. The exact solutions are crucial in theoretical mathematics, while the approximate solutions are often more practical in real-world applications. In conclusion, we’ve successfully navigated through completing the square and found the solutions to our quadratic equation.

Why Completing the Square Matters

You might be thinking, “Okay, we solved it, but why bother with completing the square when we have other methods like the quadratic formula?” Great question! Completing the square is more than just a method for solving quadratic equations. It's a fundamental technique that underpins many concepts in algebra and calculus. It helps in:

• Deriving the Quadratic Formula: The quadratic formula itself is derived by completing the square on the general quadratic equation $ax^2 + bx + c = 0$.
• Finding the Vertex of a Parabola: As we mentioned earlier, completing the square puts the quadratic equation in vertex form, directly revealing the vertex of the parabola.
• Integrating Certain Functions in Calculus: Completing the square is often used as a step in integrating rational functions.
• Understanding Transformations of Functions: It provides insights into how changing the parameters of a quadratic function affects its graph.

Completing the square enhances your understanding of the structure and properties of quadratic equations, making it a valuable tool in your mathematical toolkit. It's a versatile technique that extends beyond simply finding solutions, offering a deeper appreciation for mathematical relationships and transformations. By mastering completing the square, you're not just learning a method; you're building a foundational understanding that will serve you well in more advanced mathematical studies.

Practice Makes Perfect

The best way to master completing the square is, well, to practice! Try solving different quadratic equations using this method. Vary the coefficients, introduce fractions, and challenge yourself with more complex problems. Each problem you solve will reinforce your understanding and make the process more intuitive. Don't get discouraged if you stumble at first; completing the square takes practice. The key is to break down each problem into steps, follow the procedure meticulously, and learn from any mistakes. Consistent practice will build your confidence and solidify your skills. Remember, every mathematical technique becomes easier with repetition. So, grab a pencil, find some practice problems, and start squaring those equations! You’ve got this!

Conclusion

So, there you have it! We've successfully solved the quadratic equation $-3g^2 + 12g = -57$ by completing the square. Remember the steps: make the leading coefficient 1, complete the square, factor and simplify, take the square root, and solve for g. It might seem like a lot of steps, but with practice, it becomes second nature. And remember, completing the square isn't just about finding solutions; it's about understanding the underlying structure of quadratic equations and strengthening your problem-solving skills. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Cheers to mastering another crucial algebraic technique! You've taken a significant step in enhancing your mathematical abilities, and I'm confident that you'll continue to excel in your mathematical journey.