Solving (x-10)(x-6)=-4 Using The Quadratic Formula

In the realm of mathematics, quadratic equations hold a significant place. These equations, characterized by the presence of a variable raised to the power of two, appear in various fields, from physics and engineering to economics and computer science. Among the methods available for solving quadratic equations, the quadratic formula stands out as a versatile and reliable tool.

Understanding Quadratic Equations

Before delving into the application of the quadratic formula, it's crucial to grasp the fundamental nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning that the highest power of the variable is two. The general form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where:

• x represents the variable we aim to solve for.
• a, b, and c are constants, with a not equal to zero (otherwise, the equation would be linear).

The coefficients a, b, and c play a vital role in determining the characteristics and solutions of the quadratic equation. The term $ax^2$ is the quadratic term, $bx$ is the linear term, and $c$ is the constant term.

Quadratic equations can have up to two solutions, also known as roots or zeros. These solutions represent the values of x that satisfy the equation, meaning that when substituted into the equation, they make the equation true. Geometrically, the solutions correspond to the points where the graph of the quadratic equation (a parabola) intersects the x-axis.

The Quadratic Formula: A Universal Solution

The quadratic formula provides a direct method for finding the solutions of any quadratic equation, regardless of its complexity. It is a powerful tool that eliminates the need for factoring or completing the square, especially when dealing with equations that are difficult or impossible to solve using these methods. The quadratic formula is expressed as follows:

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

This formula reveals that the solutions of a quadratic equation depend on the coefficients a, b, and c. The expression under the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the solutions:

• If $b^2 - 4ac > 0$, the equation has two distinct real solutions.
• If $b^2 - 4ac = 0$, the equation has one real solution (a repeated root).
• If $b^2 - 4ac < 0$, the equation has two complex solutions.

Applying the Quadratic Formula: A Step-by-Step Guide

To effectively employ the quadratic formula, follow these steps:

1. Rearrange the Equation: Ensure the quadratic equation is in the standard form $ax^2 + bx + c = 0$. If necessary, rearrange the terms to match this form.
2. Identify Coefficients: Determine the values of the coefficients a, b, and c from the standard form equation.
3. Calculate the Discriminant: Compute the discriminant, $b^2 - 4ac$, to ascertain the nature of the solutions.
4. Apply the Quadratic Formula: Substitute the values of a, b, and c into the quadratic formula and simplify the expression.
5. Find the Solutions: Solve the equation obtained in the previous step to find the two possible solutions for x.

Example: Solving $(x-10)(x-6) = -4$

Let's illustrate the application of the quadratic formula by solving the equation $(x-10)(x-6) = -4$.

Step 1: Rearrange the Equation

First, expand the left side of the equation:

$x^2 - 16x + 60 = -4$

Next, add 4 to both sides to bring the equation to standard form:

$x^2 - 16x + 64 = 0$

Step 2: Identify Coefficients

Now, identify the coefficients:

• a = 1
• b = -16
• c = 64

Step 3: Calculate the Discriminant

Compute the discriminant:

$b^2 - 4ac = (-16)^2 - 4(1)(64) = 256 - 256 = 0$

The discriminant is zero, indicating that the equation has one real solution (a repeated root).

Step 4: Apply the Quadratic Formula

Substitute the values of a, b, and c into the quadratic formula:

$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(64)}}{2(1)}$

Simplify the expression:

$x = \frac{16 \pm \sqrt{0}}{2}$

$x = \frac{16}{2}$

Step 5: Find the Solutions

Solve for x:

$x = 8$

Therefore, the equation $(x-10)(x-6) = -4$ has one real solution: x = 8.

Advantages of the Quadratic Formula

The quadratic formula offers several advantages over other methods for solving quadratic equations:

• Universality: It can be applied to any quadratic equation, regardless of its complexity or the nature of its coefficients.
• Direct Solution: It provides a direct method for finding the solutions without requiring factoring or completing the square.
• Discriminant Insight: The discriminant reveals the nature of the solutions (real, complex, distinct, or repeated) before actually solving the equation.
• Efficiency: It is a relatively efficient method, especially for equations that are difficult to factor or complete the square.

Conclusion

The quadratic formula stands as a cornerstone in the realm of quadratic equations. Its ability to solve any quadratic equation, coupled with its insights into the nature of solutions, makes it an indispensable tool for mathematicians, scientists, and engineers alike. By mastering the application of the quadratic formula, one gains a powerful technique for tackling a wide range of mathematical problems and real-world applications.

Understanding and utilizing the quadratic formula not only empowers individuals to solve mathematical problems but also enhances their problem-solving abilities in various domains. Its significance extends beyond the classroom, finding practical applications in diverse fields, such as physics, engineering, and finance. The quadratic formula serves as a testament to the elegance and power of mathematical tools in unraveling complex problems and advancing our understanding of the world around us.

In conclusion, the quadratic formula is a fundamental concept in mathematics that provides a systematic approach to solving quadratic equations. Its versatility, efficiency, and the insights it offers into the nature of solutions make it an invaluable tool for students, educators, and professionals alike. By mastering the quadratic formula, individuals gain a powerful mathematical skill that can be applied to a wide range of problems and real-world scenarios.