Solving 4x² + 4x - 3 = 0 Finding The Values Of X
Unlocking the solutions to quadratic equations is a fundamental skill in mathematics. In this article, we will explore the process of finding the values of x that satisfy the equation 4x² + 4x - 3 = 0. This equation, a classic example of a quadratic equation, can be solved using various methods, each offering a unique perspective on the problem. We will delve into factoring, the quadratic formula, and completing the square, illustrating each approach with detailed steps and explanations. Understanding these methods will not only equip you to solve this particular equation but also provide a solid foundation for tackling a wide range of quadratic equations.
Understanding Quadratic Equations
Before diving into the solution, let's first grasp the essence of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (x in this 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 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that make the equation true. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots depends on the discriminant, which is given by the expression b² - 4ac. If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root; and if it's negative, there are two complex roots.
In our specific equation, 4x² + 4x - 3 = 0, we can identify the coefficients as a = 4, b = 4, and c = -3. These coefficients play a crucial role in determining the roots of the equation using different methods. For instance, in factoring, we look for two numbers that multiply to ac and add up to b. In the quadratic formula, these coefficients are directly plugged into a formula to calculate the roots. In completing the square, we manipulate the equation using these coefficients to rewrite it in a form that allows us to easily solve for x. Each method offers a unique pathway to the solution, and understanding the interplay of these coefficients is key to mastering quadratic equations.
Method 1: Factoring the Quadratic Equation
Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as the product of two linear factors. The basic idea behind factoring is to reverse the process of expanding two binomials. In other words, we aim to rewrite the quadratic expression 4x² + 4x - 3 as (px + q)(rx + s), where p, q, r, and s are constants. To achieve this, we need to find two numbers that multiply to ac (4 * -3 = -12) and add up to b (4). These numbers are 6 and -2, since 6 * -2 = -12 and 6 + (-2) = 4.
Now, we can rewrite the middle term of the equation using these two numbers: 4x² + 6x - 2x - 3 = 0. This step is crucial as it allows us to group the terms and factor by grouping. We group the first two terms and the last two terms: (4x² + 6x) + (-2x - 3) = 0. From the first group, we can factor out 2x, and from the second group, we can factor out -1: 2x(2x + 3) - 1(2x + 3) = 0. Notice that we now have a common factor of (2x + 3) in both terms. We can factor this out to obtain (2x + 3)(2x - 1) = 0. This is the factored form of the quadratic equation.
To find the values of x that satisfy the equation, we set each factor equal to zero: 2x + 3 = 0 and 2x - 1 = 0. Solving the first equation, we subtract 3 from both sides and then divide by 2, giving us x = -3/2. Solving the second equation, we add 1 to both sides and then divide by 2, giving us x = 1/2. Therefore, the values of x that satisfy the equation 4x² + 4x - 3 = 0 are x = -3/2 and x = 1/2. Factoring provides a clear and intuitive way to find the roots of the equation, especially when the coefficients allow for straightforward factorization.
Method 2: Applying the Quadratic Formula
The quadratic formula is a universally applicable method for solving quadratic equations, regardless of whether they can be easily factored. It provides a direct way to calculate the roots of any quadratic equation in the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
This formula elegantly encapsulates the relationship between the coefficients of the quadratic equation and its roots. The ± sign indicates that there are two possible solutions, corresponding to the two roots of the equation.
To apply the quadratic formula to our equation, 4x² + 4x - 3 = 0, we first identify the coefficients: a = 4, b = 4, and c = -3. Now, we substitute these values into the quadratic formula:
x = (-4 ± √(4² - 4 * 4 * -3)) / (2 * 4)
Next, we simplify the expression under the square root: 4² - 4 * 4 * -3 = 16 + 48 = 64. So, the equation becomes:
x = (-4 ± √64) / 8
The square root of 64 is 8, so we have:
x = (-4 ± 8) / 8
Now, we consider the two possible solutions, one with the plus sign and one with the minus sign. For the plus sign:
x = (-4 + 8) / 8 = 4 / 8 = 1/2
For the minus sign:
x = (-4 - 8) / 8 = -12 / 8 = -3/2
Therefore, the values of x that satisfy the equation 4x² + 4x - 3 = 0, as determined by the quadratic formula, are x = 1/2 and x = -3/2. This result aligns perfectly with the solutions we obtained through factoring. The quadratic formula serves as a reliable tool, especially when factoring proves challenging or impossible. Its strength lies in its ability to handle any quadratic equation, regardless of the complexity of its coefficients.
Method 3: Completing the Square
Completing the square is another powerful method for solving quadratic equations. It involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. This method provides a deeper understanding of the structure of quadratic equations and their solutions. To apply completing the square to the equation 4x² + 4x - 3 = 0, we first divide the entire equation by the coefficient of x², which is 4, to make the coefficient of x² equal to 1:
x² + x - 3/4 = 0
Next, we move the constant term to the right side of the equation:
x² + x = 3/4
Now, we complete the square on the left side. To do this, we take half of the coefficient of the x term (which is 1), square it (1/2)² = 1/4, and add it to both sides of the equation:
x² + x + 1/4 = 3/4 + 1/4
The left side is now a perfect square trinomial, which can be factored as (x + 1/2)²:
(x + 1/2)² = 1
Now, we take the square root of both sides:
x + 1/2 = ±√1
x + 1/2 = ±1
We have two possible solutions, one with the plus sign and one with the minus sign. For the plus sign:
x + 1/2 = 1
x = 1 - 1/2
x = 1/2
For the minus sign:
x + 1/2 = -1
x = -1 - 1/2
x = -3/2
Therefore, the values of x that satisfy the equation 4x² + 4x - 3 = 0, as determined by completing the square, are x = 1/2 and x = -3/2. This result is consistent with the solutions we found using factoring and the quadratic formula. Completing the square is a valuable method for solving quadratic equations and also forms the basis for deriving the quadratic formula itself. It showcases the underlying structure of quadratic equations and provides a systematic approach to finding their roots.
Conclusion
In this comprehensive exploration, we've successfully determined the values of x in the equation 4x² + 4x - 3 = 0 using three distinct methods: factoring, the quadratic formula, and completing the square. Each method offered a unique pathway to the solution, highlighting different aspects of quadratic equations. Factoring, when applicable, provides an elegant and intuitive approach. The quadratic formula stands as a universal tool, capable of solving any quadratic equation. Completing the square offers a deeper understanding of the equation's structure and serves as the foundation for the quadratic formula itself.
The solutions we obtained through each method, x = 1/2 and x = -3/2, consistently demonstrate the versatility and reliability of these techniques. Mastering these methods empowers you to confidently tackle a wide array of quadratic equations and lays a strong foundation for further mathematical explorations. Understanding quadratic equations is crucial in various fields, including physics, engineering, and computer science. The ability to solve them efficiently and accurately is a valuable skill that will serve you well in both academic and practical settings. By mastering these techniques, you gain not only the ability to find solutions but also a deeper appreciation for the elegance and power of mathematics.
