Solving Quadratic Equations: Inga's Approach

Hey guys! Ever wondered how to crack a quadratic equation? Let's dive into how Inga tackles $2 x^2+12 x-3=0$. We'll break down the steps and see what choices could work. Solving quadratic equations is a fundamental skill in algebra, and understanding the different methods is super important. We'll explore completing the square in this case, a powerful technique that transforms the equation into a form that's easier to solve. It's like a mathematical puzzle, and Inga's approach gives us a clear path to the solution. Get ready to learn some cool tricks and understand the logic behind each step. Let's get started and see what Inga is up to! In this comprehensive guide, we'll examine each option and determine the correct steps Inga could use to solve the given quadratic equation. This exploration will not only help you understand the solution but also reinforce your understanding of completing the square and manipulating quadratic equations. Understanding these steps is crucial for mastering quadratic equations and related concepts. Let's analyze Inga's potential moves. First, the original equation $2 x^2+12 x-3=0$ needs to be rearranged to isolate the terms involving 'x'. We want to complete the square, which means we aim to create a perfect square trinomial. This is a trinomial that can be factored into the square of a binomial, making the equation easier to solve. The process involves several key steps: factoring, adding a constant, and simplifying. Remember, the goal is to transform the equation into a form where we can easily extract the values of 'x'. So, let's look at each option and see how it aligns with the correct method.

Analyzing Inga's Steps: A Detailed Breakdown

Let's meticulously analyze the given options to see which steps align with the correct method for solving the quadratic equation. This will involve understanding the principles of completing the square and how to apply them. It's all about rearranging and manipulating the equation to arrive at a solvable form. We are looking for steps that follow the logical progression of completing the square. This includes isolating the 'x' terms, factoring, and adding a constant to both sides of the equation. Each step is essential, and understanding their purpose is crucial to solving the equation. Remember, completing the square is a process of transforming a quadratic expression into a perfect square trinomial plus a constant. This technique helps to easily solve for 'x'. The process requires careful attention to detail, and a good grasp of algebraic manipulation is key. Remember the initial goal of isolating 'x' and turning it into a solvable form. Let's break down each option now. We will check whether each step is valid, invalid, or partially correct, so we'll be able to understand the process well. The proper application of these steps will not only allow us to get the correct answer but also give us a great understanding of the underlying principles.

Option A: $2\left(x^2+6 x+9\right)=3+18$

Let's break down option A: $2\left(x^2+6 x+9\right)=3+18$. Does this step make sense in Inga's quest to solve the quadratic equation? This option appears to be a good move in completing the square. First, Inga would factor out the 2 from the x terms, which gives us $2(x^2+6x)$. To complete the square inside the parenthesis, Inga needs to add a constant. This constant comes from taking half of the coefficient of the x term (which is 6), squaring it (3^2 = 9), and adding it inside the parenthesis. But we need to keep the equation balanced. Since there's a 2 outside the parenthesis, Inga actually added 2*9 = 18 to the left side. So, to keep things balanced, she adds 18 to the right side, too. So, if Inga started with $2 x^2+12 x-3=0$, moving the -3 to the right side gives $2 x^2+12 x=3$. Factoring the 2 gives $2(x^2+6x)=3$. Adding 9 inside the parenthesis (and 18 to the right side) is the correct next step. Therefore, this option is valid. By completing the square, Inga is essentially rewriting the quadratic expression in a way that simplifies the process of finding the roots. This method involves manipulating the equation to create a perfect square trinomial on one side. This is crucial as it allows us to express the quadratic equation in a form that makes the solution more straightforward. Option A seems to have performed this step correctly. So, based on our analysis, we can confidently say that option A is a valid step. Completing the square is not just about finding the right answer; it's about understanding the underlying structure of quadratic equations and how they can be manipulated to reveal their solutions. It is about understanding the logic behind mathematical operations. The next step would be to simplify the equation, taking the square root of both sides, and then solving for x. So, Option A is a solid choice in the journey to solve the equation.

Option B: $2\left(x^2+6 x\right)=-3$

Now, let's examine option B: $2\left(x^2+6 x\right)=-3$. This is not the correct step to solve the quadratic equation $2 x^2+12 x-3=0$. In this, Inga started with the original equation $2 x^2+12 x-3=0$ and moved -3 to the right side of the equation, which gives $2 x^2+12 x=3$. She also correctly factored out the 2. The proper next step would be to complete the square by adding a constant term inside the parenthesis. This step is a necessary intermediate step in the process of completing the square. Inga should add a constant term inside the parentheses to create a perfect square trinomial. However, according to option B, it skips the step of adding a constant to both sides of the equation, which is essential to keep the equation balanced while completing the square. Inga missed the important step of adding a constant term to both sides of the equation, which will not lead to a correct answer. Therefore, this option is invalid. Completing the square is a fundamental technique for solving quadratic equations, and understanding its steps is crucial. This step is about manipulating the equation to create a perfect square trinomial on one side and then solving for 'x'. Option B does not properly do so; this is why it is not a valid approach.

Option C: $2\left(x^2+6 x\right)=3$

Let's break down option C: $2\left(x^2+6 x\right)=3$. This is a crucial step in the process of solving the quadratic equation $2 x^2+12 x-3=0$. Option C starts by rearranging the original equation. First, Inga would have to move -3 to the right side, giving $2 x^2+12 x=3$. Next, she factors out the 2 from the x terms, resulting in $2(x^2+6x)=3$. This is indeed a valid step in solving the equation. The next step would be to complete the square. By completing the square, Inga is getting the equation ready for the next steps in solving for x. Remember, the goal is to transform the equation into a form that's easier to solve. The process involves several key steps: factoring, adding a constant, and simplifying. This option sets up the equation for completing the square, a key technique for solving the equation. So, if we look at the original equation $2 x^2+12 x-3=0$, then adding 3 to both sides to isolate the 'x' terms, and factoring the 2 from the left side, we get exactly the equation in option C. Therefore, this is a correct step. Thus, Option C is a valid and necessary step. By understanding how to manipulate and transform quadratic equations, Inga can find the roots of the equation.

Option D: $x+3= \pmDiscussion category : mathematics$

Finally, let's examine option D: $x+3= \pmDiscussion category : mathematics$. This option seems to be incorrect, because it appears to be a solution to the equation, and it should only appear at the end. Inga would need to complete the square, and she did not follow the steps properly, so this option is invalid. It does not follow from the initial equation. It suggests that Inga had already completed the square, simplified the equation, and taken the square root of both sides. However, the correct steps to complete the square were not taken in the previous options. The correct approach involves isolating the 'x' terms, factoring out the leading coefficient if necessary, completing the square, and then solving for 'x'. Option D seems to skip these crucial steps and jumps directly to a final result, without the intermediate steps. In the process of solving the equation, Inga needs to manipulate and transform the equation to make it into a solvable format. This involves a series of steps that build upon each other. So, this option is not valid, as it does not follow the correct steps to solve the equation. Therefore, this option is invalid.

Conclusion: Selecting the Right Steps

So, guys, after analyzing each option, it's clear which steps Inga could have used to solve the quadratic equation $2 x^2+12 x-3=0$. Option A and C provide correct steps to solve the quadratic equation using the method of completing the square. Remember, when solving these types of equations, understanding each step is super important. We hope this explanation helps you understand how Inga can solve this equation! Keep practicing, and you'll become a quadratic equation master in no time!