Solving X^2 + 20x + 100 = 36 A Step By Step Guide
In the realm of mathematics, solving quadratic equations is a fundamental skill. The equation presented, x^2 + 20x + 100 = 36, is a classic example of a quadratic equation that can be solved using various methods. This article aims to provide a step-by-step guide on how to solve this equation, ensuring a clear understanding of the underlying concepts and techniques. We will delve into the process of simplifying the equation, identifying the most appropriate method for solving it, and arriving at the correct solutions for x. Whether you're a student grappling with algebra or someone looking to refresh your math skills, this guide will offer a comprehensive approach to tackling quadratic equations.
Understanding Quadratic Equations
Before diving into the solution, let's first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠0. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of x that satisfy the equation. In the given equation, x^2 + 20x + 100 = 36, we can identify a = 1, b = 20, and c is a value we need to determine after rearranging the equation. Recognizing the structure of a quadratic equation is the first step toward solving it effectively.
Methods for Solving Quadratic Equations
There are several methods for solving quadratic equations, each with its own advantages and suitability depending on the specific equation. The most common methods include:
- Factoring: This method involves expressing the quadratic equation as a product of two binomials. It is often the quickest method when the equation can be easily factored.
- Completing the Square: This method involves manipulating the equation to form a perfect square trinomial, which can then be easily solved.
- Quadratic Formula: This is a general formula that can be used to solve any quadratic equation. It is particularly useful when the equation is difficult to factor.
For the equation x^2 + 20x + 100 = 36, we will explore the most efficient method to find the values of x that satisfy the equation. By understanding these methods, you can choose the most appropriate one for any given quadratic equation, making problem-solving more efficient and accurate.
Step-by-Step Solution
Now, let's solve the equation x^2 + 20x + 100 = 36 step by step. The goal is to isolate x and find its possible values. We will use a combination of algebraic manipulation and factoring techniques to arrive at the solution.
Step 1: Simplify the Equation
The first step is to simplify the equation by moving all terms to one side, setting the equation equal to zero. This is a standard practice for solving quadratic equations, as it allows us to apply factoring or the quadratic formula more easily. Subtract 36 from both sides of the equation:
x^2 + 20x + 100 - 36 = 36 - 36
This simplifies to:
x^2 + 20x + 64 = 0
Now we have a standard form quadratic equation, where a = 1, b = 20, and c = 64. This simplified form is crucial for the next steps in solving the equation. By setting the equation to zero, we create a clear framework for finding the roots.
Step 2: Factor the Quadratic Equation
The next step is to factor the quadratic equation. Factoring involves finding two binomials that, when multiplied together, give the quadratic expression. This method is efficient when the quadratic expression can be easily factored. We are looking for two numbers that multiply to 64 and add up to 20. Consider the factors of 64:
- 1 and 64
- 2 and 32
- 4 and 16
- 8 and 8
The pair 4 and 16 add up to 20, so we can factor the quadratic expression as follows:
(x + 4)(x + 16) = 0
This factorization is a critical step. By expressing the quadratic equation as a product of two binomials, we can easily find the values of x that make the equation true. Each binomial represents a potential solution for x.
Step 3: Solve for x
Now that we have factored the equation, we can solve for x. The principle here is that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:
- x + 4 = 0
- x + 16 = 0
Solving the first equation:
x + 4 = 0 x = -4
Solving the second equation:
x + 16 = 0 x = -16
Therefore, the solutions for x are -4 and -16. These are the values that, when substituted back into the original equation, will make the equation true. This step is the culmination of the factoring process, providing the specific values of x that satisfy the quadratic equation.
Verifying the Solutions
To ensure the accuracy of our solutions, it's essential to verify them by substituting them back into the original equation. This process confirms that the values we found for x indeed satisfy the equation. Let's verify the solutions x = -4 and x = -16.
Verification for x = -4
Substitute x = -4 into the original equation:
(-4)^2 + 20(-4) + 100 = 36
Simplify the equation:
16 - 80 + 100 = 36
Combine the terms:
36 = 36
Since the equation holds true, x = -4 is a valid solution. This verification step confirms that our solution is accurate and consistent with the original equation.
Verification for x = -16
Now, substitute x = -16 into the original equation:
(-16)^2 + 20(-16) + 100 = 36
Simplify the equation:
256 - 320 + 100 = 36
Combine the terms:
36 = 36
Since the equation holds true, x = -16 is also a valid solution. This second verification further solidifies the accuracy of our solutions, demonstrating that both x = -4 and x = -16 satisfy the original quadratic equation.
Conclusion
In conclusion, the solutions for x in the equation x^2 + 20x + 100 = 36 are x = -4 and x = -16. We arrived at these solutions by simplifying the equation, factoring the quadratic expression, and then solving for x. We also verified the solutions by substituting them back into the original equation, confirming their accuracy.
Key Takeaways
- Solving quadratic equations often involves simplifying the equation and then factoring or using the quadratic formula.
- Factoring is an efficient method when the quadratic expression can be easily factored.
- Verifying solutions is crucial to ensure accuracy.
Understanding how to solve quadratic equations is a fundamental skill in mathematics. This step-by-step guide has provided a comprehensive approach to solving the equation x^2 + 20x + 100 = 36. By mastering these techniques, you can confidently tackle similar problems and deepen your understanding of algebra. Remember to practice regularly and apply these methods to various quadratic equations to strengthen your problem-solving skills. The ability to solve quadratic equations is not only essential for academic success but also has practical applications in various fields, making it a valuable skill to acquire.