Solving Quadratic Equations Inga's Approach To 2x^2 + 12x - 3 = 0
Solving quadratic equations is a fundamental skill in algebra, and understanding different methods to tackle these equations is crucial. In this comprehensive guide, we'll delve into Inga's approach to solving the quadratic equation 2x^2 + 12x - 3 = 0. We will explore the steps she could take, analyzing the correctness of each option, and providing a clear understanding of the underlying mathematical principles. Mastering these techniques will empower you to confidently solve a wide range of quadratic equations and appreciate their applications in various fields.
Understanding Quadratic Equations
Before diving into Inga's solution, let's establish a solid foundation by defining what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠0. The solutions to a quadratic equation, also known as roots or zeros, represent the values of x that satisfy the equation. These roots can be real or complex numbers, and a quadratic equation can have up to two distinct roots.
There are several methods to solve quadratic equations, each with its own advantages and applicability. Some common methods include factoring, completing the square, and using the quadratic formula. Factoring involves expressing the quadratic expression as a product of two linear factors, while completing the square transforms the equation into a perfect square trinomial. The quadratic formula provides a direct solution for any quadratic equation, regardless of its factorability.
The discriminant, denoted as Δ = b^2 - 4ac, plays a crucial role in determining the nature of the roots. If Δ > 0, the equation has two distinct real roots; if Δ = 0, it has one real root (a repeated root); and if Δ < 0, it has two complex roots. Understanding the discriminant helps us predict the type of solutions we can expect before embarking on the solving process.
Analyzing Inga's Steps for Solving 2x^2 + 12x - 3 = 0
Now, let's focus on the specific quadratic equation Inga is trying to solve: 2x^2 + 12x - 3 = 0. We'll examine the steps she could use to find the solutions, carefully evaluating each option to determine its validity and effectiveness.
Option 1: 2(x^2 + 6x + 9) = -3 + 9
This step represents an attempt to complete the square. To analyze its correctness, let's break it down. First, the equation 2x^2 + 12x - 3 = 0 is divided by 2 (only conceptually for now) resulting in x^2 + 6x - 3/2 = 0. The next logical step in completing the square is to add (b/2)^2 to both sides of the equation, where b is the coefficient of the x term. In this case, b = 6, so (b/2)^2 = (6/2)^2 = 3^2 = 9. However, the important thing here is that the original equation was not divided by 2 on both sides. So, inside the parenthesis, we are adding 9, which is then multiplied by 2, to the left side of the equation. So, to balance the equation, we need to add 2 * 9 = 18 to the right side. The right side of the equation should therefore be -3 + 18 = 15, not -3 + 9 = 6.
Hence, the equation 2(x^2 + 6x + 9) = -3 + 9 is incorrect because it doesn't properly balance the equation after completing the square.
Option 2: x + 3 = ±√(21/2)
This step seems to follow from completing the square, but its correctness depends on the preceding steps. Let's assume for a moment that we had correctly completed the square and arrived at an equation of the form 2(x + 3)^2 = C, where C is some constant. Dividing both sides by 2, we get (x + 3)^2 = C/2. Taking the square root of both sides yields x + 3 = ±√(C/2).
To determine if x + 3 = ±√(21/2) is correct, we need to trace back the steps. If C were 21, then this step would be valid. However, based on our analysis of Option 1, the constant term after correctly completing the square is not likely to result in 21 under the square root. So, let's verify this by completing the square correctly for the original equation.
Starting with 2x^2 + 12x - 3 = 0, we factor out the 2 from the x^2 and x terms: 2(x^2 + 6x) - 3 = 0. To complete the square inside the parentheses, we add and subtract (6/2)^2 = 9: 2(x^2 + 6x + 9 - 9) - 3 = 0. This gives us 2((x + 3)^2 - 9) - 3 = 0. Expanding, we get 2(x + 3)^2 - 18 - 3 = 0, which simplifies to 2(x + 3)^2 = 21. Dividing by 2, we have (x + 3)^2 = 21/2. Taking the square root, we arrive at x + 3 = ±√(21/2).
Therefore, the equation x + 3 = ±√(21/2) is correct as a step in solving the quadratic equation by completing the square.
Option 3: 2(x^2 + 6x) = 3
This step involves isolating the terms with x on one side of the equation. Starting with 2x^2 + 12x - 3 = 0, we can add 3 to both sides to get 2x^2 + 12x = 3. Factoring out a 2 from the left side, we obtain 2(x^2 + 6x) = 3.
Thus, the equation 2(x^2 + 6x) = 3 is a correct step in the process of solving the quadratic equation, specifically as an initial step towards completing the square.
Option 4: 2(x^2 + 6x + 9) = 3
This step attempts to complete the square within the parentheses. However, it makes a critical error in balancing the equation. From the previous step, we have 2(x^2 + 6x) = 3. To complete the square inside the parentheses, we need to add (6/2)^2 = 9. This gives us 2(x^2 + 6x + 9) on the left side. However, since we're adding 9 inside the parentheses, which is then multiplied by 2, we're effectively adding 18 to the left side. To maintain balance, we must add 18 to the right side as well. Thus, the correct equation should be 2(x^2 + 6x + 9) = 3 + 18 = 21, not 2(x^2 + 6x + 9) = 3.
Therefore, the equation 2(x^2 + 6x + 9) = 3 is incorrect due to the failure to properly balance the equation after completing the square.
Choosing the Correct Answers
Based on our analysis, the three correct steps Inga could use to solve the quadratic equation 2x^2 + 12x - 3 = 0 are:
- Option 2: x + 3 = ±√(21/2)
- Option 3: 2(x^2 + 6x) = 3
Alternative Methods for Solving Quadratic Equations
While Inga's approach focuses on completing the square, it's essential to recognize that there are alternative methods for solving quadratic equations. Let's briefly explore two other common techniques:
Factoring
Factoring involves expressing the quadratic expression as a product of two linear factors. This method is most effective when the quadratic expression can be easily factored. For example, the equation x^2 + 5x + 6 = 0 can be factored as (x + 2)(x + 3) = 0, leading to the solutions x = -2 and x = -3. However, many quadratic equations are not easily factorable, making this method less versatile than others.
Quadratic Formula
The quadratic formula provides a universal solution for any quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
This formula directly yields the roots of the equation, regardless of its factorability. Applying the quadratic formula to 2x^2 + 12x - 3 = 0, where a = 2, b = 12, and c = -3, we get:
x = (-12 ± √(12^2 - 4 * 2 * -3)) / (2 * 2)
x = (-12 ± √(144 + 24)) / 4
x = (-12 ± √168) / 4
x = (-12 ± 2√42) / 4
x = (-6 ± √42) / 2
Thus, the solutions are x = (-6 + √42) / 2 and x = (-6 - √42) / 2.
Conclusion
Solving quadratic equations is a fundamental skill with various applications in mathematics and other disciplines. Inga's approach, focusing on completing the square, provides a valuable method for finding the solutions. By carefully analyzing the steps and understanding the underlying principles, we can confidently solve a wide range of quadratic equations. Additionally, recognizing alternative methods like factoring and the quadratic formula expands our problem-solving toolkit. The correct steps Inga could use to solve 2x^2 + 12x - 3 = 0 are x + 3 = ±√(21/2) and 2(x^2 + 6x) = 3. By mastering these techniques, you'll be well-equipped to tackle quadratic equations and appreciate their significance in mathematical problem-solving.
