Solving $x^2 - 10x + 16 = 0$ A Comprehensive Guide

Jul 10, 2025 by ADMIN 51 views

In this article, we will delve into the process of solving the quadratic equation $x^2 - 10x + 16 = 0$ . Quadratic equations are fundamental in algebra and have numerous applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill in mathematics. We will explore different methods to find the roots (or solutions) of this equation, including factoring, completing the square, and using the quadratic formula. Each method provides a unique approach, and choosing the most efficient one often depends on the specific form of the equation.

Method 1: Factoring

Factoring is often the quickest method for solving quadratic equations when it is applicable. This method involves expressing the quadratic expression as a product of two binomials. For the given equation, $x^2 - 10x + 16 = 0$ , we need to find two numbers that multiply to 16 (the constant term) and add up to -10 (the coefficient of the x term). These two numbers are -2 and -8 because $(-2) imes (-8) = 16$ and $(-2) + (-8) = -10$ . Thus, we can rewrite the equation as follows:

$x^2 - 10x + 16 = (x - 2)(x - 8) = 0$

Now, according to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations:

$x - 2 = 0$
$x - 8 = 0$

Solving these equations for x, we get:

$x = 2$
$x = 8$

Therefore, the solutions to the equation $x^2 - 10x + 16 = 0$ are $x = 2$ and $x = 8$ . Factoring is an efficient method when the quadratic expression can be easily factored, and it provides a straightforward way to find the roots of the equation.

Method 2: Completing the Square

Completing the square is another powerful method for solving quadratic equations. This technique involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. For the equation $x^2 - 10x + 16 = 0$ , we start by isolating the constant term:

$x^2 - 10x = -16$

Next, we need to add a value to both sides of the equation to complete the square. This value is determined by taking half of the coefficient of the x term (which is -10), squaring it, and adding the result to both sides. Half of -10 is -5, and $(-5)^2$ is 25. So, we add 25 to both sides:

$x^2 - 10x + 25 = -16 + 25$

Now, the left side of the equation is a perfect square trinomial, which can be factored as $(x - 5)^2$ . The right side simplifies to 9:

$(x - 5)^2 = 9$

To solve for x, we take the square root of both sides:

$x - 5 = \pm\sqrt{9}$

$x - 5 = \pm 3$

This gives us two separate equations:

$x - 5 = 3$
$x - 5 = -3$

Solving these equations for x, we get:

$x = 3 + 5 = 8$
$x = -3 + 5 = 2$

Thus, the solutions to the equation $x^2 - 10x + 16 = 0$ are $x = 2$ and $x = 8$ , which matches the solutions we found by factoring. Completing the square is particularly useful when the quadratic equation is not easily factorable, and it provides a systematic approach to finding the solutions.

Method 3: Quadratic Formula

The quadratic formula is a universally applicable method for solving quadratic equations of the form $ax^2 + bx + c = 0$ . The formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 - 10x + 16 = 0$ , we have $a = 1$ , $b = -10$ , and $c = 16$ . Plugging these values into the quadratic formula, we get:

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(16)}}{2(1)}$

$x = \frac{10 \pm \sqrt{100 - 64}}{2}$

$x = \frac{10 \pm \sqrt{36}}{2}$

$x = \frac{10 \pm 6}{2}$

This gives us two possible solutions:

$x = \frac{10 + 6}{2} = \frac{16}{2} = 8$
$x = \frac{10 - 6}{2} = \frac{4}{2} = 2$

Therefore, the solutions to the equation $x^2 - 10x + 16 = 0$ are $x = 2$ and $x = 8$ , consistent with the solutions found using factoring and completing the square. The quadratic formula is a powerful tool because it can be used to solve any quadratic equation, regardless of whether it can be easily factored or not.

Comparison of Methods

Each method we've discussed—factoring, completing the square, and the quadratic formula—has its strengths and weaknesses. Factoring is generally the quickest method when the quadratic expression can be easily factored. It relies on recognizing the factors that satisfy the equation and is often the preferred method for simple quadratic equations. However, not all quadratic equations are easily factorable, which limits the applicability of this method.

Completing the square is a more systematic approach that can be used even when factoring is not straightforward. It involves transforming the equation into a perfect square trinomial, which then allows for easy solution. This method is particularly useful for understanding the structure of quadratic equations and is a foundational technique for deriving the quadratic formula. However, completing the square can be more time-consuming than factoring, especially when the coefficients are not integers.

The quadratic formula is the most versatile method, as it can be used to solve any quadratic equation. It provides a direct way to find the solutions by plugging the coefficients of the equation into the formula. While it may seem more complex than factoring, it guarantees a solution, even when the roots are irrational or complex. The quadratic formula is an essential tool in algebra and is widely used in various mathematical and scientific applications.

Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications. In physics, they are used to model projectile motion, where the path of an object thrown into the air can be described by a quadratic equation. The height of the object at any given time can be determined by solving a quadratic equation that represents its trajectory. Similarly, in engineering, quadratic equations are used in the design of structures, such as bridges and buildings, to ensure stability and safety.

In economics, quadratic equations can be used to model cost, revenue, and profit functions. Businesses often use these models to determine the optimal pricing and production levels that will maximize their profits. For example, the demand curve for a product might be represented by a quadratic equation, and the point at which profit is maximized can be found by solving a quadratic equation.

Quadratic equations also appear in computer graphics, where they are used to create curves and surfaces. Bezier curves, which are widely used in computer-aided design (CAD) and animation, are based on quadratic and cubic equations. These curves allow designers to create smooth and visually appealing shapes.

Understanding how to solve quadratic equations is therefore essential for students and professionals in a wide range of fields. Whether it's calculating the trajectory of a projectile, designing a bridge, or modeling economic trends, quadratic equations provide a powerful tool for problem-solving.

Conclusion

In summary, we have explored three methods for solving the quadratic equation $x^2 - 10x + 16 = 0$ : factoring, completing the square, and the quadratic formula. Each method yields the same solutions, $x = 2$ and $x = 8$ , but they differ in their approach and applicability. Factoring is the quickest method when the equation is easily factorable, while completing the square provides a systematic approach suitable for any quadratic equation. The quadratic formula is the most versatile, guaranteeing a solution regardless of the equation's complexity.

Understanding these methods and their strengths and weaknesses is crucial for effectively solving quadratic equations. Furthermore, recognizing the real-world applications of quadratic equations underscores their importance in various fields, making the study of algebra not just an academic exercise but a practical skill. Whether you choose to factor, complete the square, or apply the quadratic formula, the ability to solve quadratic equations opens doors to a deeper understanding of mathematics and its applications.