Solving 4x² - 16x + 16 = 0 A Detailed Guide
This article provides a detailed walkthrough on solving the quadratic equation 4x² - 16x + 16 = 0. We will explore different methods, including factoring, using the quadratic formula, and completing the square, to find the solution set. Understanding how to solve quadratic equations is a fundamental skill in algebra, with applications in various fields like physics, engineering, and economics.
Introduction to Quadratic Equations
Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (typically 'x') 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. These equations can have up to two real solutions, also known as roots or zeros. Finding these solutions involves techniques such as factoring, completing the square, or using the quadratic formula. The nature of the solutions (real, distinct, or complex) depends on the discriminant (b² - 4ac). Quadratic equations are prevalent in many mathematical and real-world problems, from projectile motion in physics to optimization problems in economics.
In the given equation, 4x² - 16x + 16 = 0, we have a = 4, b = -16, and c = 16. Our goal is to find the values of x that satisfy this equation. We will begin by exploring the simplest method: factoring.
Method 1: Factoring
Factoring is a technique used to simplify and solve quadratic equations by expressing the quadratic expression as a product of two binomials. The idea behind factoring is to reverse the process of expanding binomials using the distributive property (also known as FOIL - First, Outer, Inner, Last). When factoring, we look for two numbers that multiply to give the constant term (c) and add up to give the coefficient of the linear term (b). If we can find such numbers, we can rewrite the quadratic equation in a factored form, making it easier to solve.
In our case, the quadratic equation is 4x² - 16x + 16 = 0. Before we start factoring, it's beneficial to look for any common factors among the coefficients. We notice that all terms are divisible by 4. Dividing the entire equation by 4 simplifies it to:
x² - 4x + 4 = 0
Now, we need to find two numbers that multiply to 4 and add up to -4. These numbers are -2 and -2. Therefore, we can rewrite the quadratic expression as:
(x - 2)(x - 2) = 0
This can also be written as:
(x - 2)² = 0
To find the solutions, we set each factor equal to zero:
x - 2 = 0
Solving for x, we get:
x = 2
Since both factors are the same, we have a repeated root. This means the quadratic equation has only one distinct solution.
Thus, by factoring, we have found that the solution to the equation 4x² - 16x + 16 = 0 is x = 2. Factoring is often the quickest method when the quadratic equation can be easily factored. However, not all quadratic equations are easily factorable. In such cases, other methods like the quadratic formula or completing the square are more appropriate.
Method 2: Quadratic Formula
The quadratic formula is a universal method for finding the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0. It is particularly useful when factoring is difficult or impossible. The formula is derived from the process of completing the square and provides a direct way to calculate the roots without needing to factor. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Where:
- a is the coefficient of the x² term
- b is the coefficient of the x term
- c is the constant term
The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant provides valuable information about the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a repeated root).
- If b² - 4ac < 0, the equation has two complex roots.
Let's apply the quadratic formula to our equation, 4x² - 16x + 16 = 0. Here, a = 4, b = -16, and c = 16. Substituting these values into the quadratic formula, we get:
x = (-(-16) ± √((-16)² - 4 * 4 * 16)) / (2 * 4)
Simplifying:
x = (16 ± √(256 - 256)) / 8
x = (16 ± √0) / 8
x = (16 ± 0) / 8
x = 16 / 8
x = 2
As we found using factoring, the quadratic formula confirms that the equation has one real solution, x = 2. The discriminant being zero indicates that the equation has a repeated root. The quadratic formula is a powerful tool that guarantees finding the solutions of any quadratic equation, regardless of its factorability. It's an essential technique in algebra and is widely used in various applications.
Method 3: Completing the Square
Completing the square is another method for solving quadratic equations, and it is especially useful when the equation is not easily factorable. This technique involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. Completing the square is not only a method for finding solutions but also a fundamental concept used in deriving the quadratic formula itself.
The general process of completing the square involves the following steps:
- Divide the entire equation by the coefficient of the x² term (if it's not 1). This makes the coefficient of x² equal to 1.
- Move the constant term to the right side of the equation.
- Take half of the coefficient of the x term, square it, and add it to both sides of the equation. This step creates a perfect square trinomial on the left side.
- Rewrite the left side as a square of a binomial.
- Take the square root of both sides of the equation.
- Solve for x.
Let's apply the method of completing the square to our equation, 4x² - 16x + 16 = 0.
First, divide the entire equation by 4 (the coefficient of x²):
x² - 4x + 4 = 0
Next, move the constant term to the right side:
x² - 4x = -4
Now, take half of the coefficient of the x term (-4), which is -2, and square it: (-2)² = 4. Add this to both sides of the equation:
x² - 4x + 4 = -4 + 4
x² - 4x + 4 = 0
The left side is now a perfect square trinomial, which can be written as:
(x - 2)² = 0
Take the square root of both sides:
√(x - 2)² = √0
x - 2 = 0
Finally, solve for x:
x = 2
Completing the square, we arrive at the same solution, x = 2. This method demonstrates how to transform a quadratic equation into a more manageable form by creating a perfect square. While it might seem more involved than factoring for simple equations, completing the square is a valuable technique for more complex quadratic equations and forms the basis for deriving the quadratic formula.
Solution Set
After applying all three methods—factoring, the quadratic formula, and completing the square—we consistently found that the solution to the quadratic equation 4x² - 16x + 16 = 0 is x = 2. Since the discriminant (b² - 4ac) is equal to zero, this indicates that the equation has one real, repeated root.
Therefore, the solution set for the equation is {2}. This means that x = 2 is the only value that satisfies the equation. Graphically, this corresponds to the parabola represented by the equation touching the x-axis at only one point, which is x = 2.
In summary, understanding the different methods for solving quadratic equations allows us to approach problems with flexibility and choose the most efficient method based on the specific equation. Factoring is often the quickest method for simple equations, while the quadratic formula and completing the square provide more robust approaches for all types of quadratic equations. The solution set is a concise way to represent all the values of x that make the equation true, in this case, only one value, x = 2.
Conclusion
In conclusion, we have successfully solved the quadratic equation 4x² - 16x + 16 = 0 using three different methods: factoring, the quadratic formula, and completing the square. Each method provided the same solution, x = 2, confirming its accuracy. The fact that the discriminant was zero indicated a single, repeated root, which we verified through our calculations.
Solving quadratic equations is a fundamental skill in algebra, and mastering these techniques is crucial for tackling more advanced mathematical problems. Factoring is often the most straightforward approach when applicable, but the quadratic formula and completing the square offer more general solutions for any quadratic equation.
The solution set, {2}, represents the value(s) of x that satisfy the given equation. In this case, only x = 2 makes the equation true. Understanding the concept of solution sets and how to find them is essential for solving not just quadratic equations but also other types of equations in mathematics.
By exploring these methods, we have gained a deeper understanding of quadratic equations and the various techniques to solve them. This knowledge is invaluable for anyone studying mathematics, science, engineering, or any field that relies on mathematical problem-solving.
