Solving Quadratic Equations 5(x+5)^2 - 38 = -18
Introduction
This article delves into the step-by-step solution of the quadratic equation 5(x+5)² - 38 = -18. Quadratic equations, characterized by the presence of a squared variable, play a crucial role in various mathematical and scientific applications. Mastering the techniques to solve these equations is essential for anyone pursuing studies or careers in STEM fields. In this guide, we will employ algebraic manipulations to isolate the variable and ultimately determine the values of x that satisfy the given equation. Our approach will emphasize clarity and precision, ensuring that each step is thoroughly explained. This comprehensive solution will not only provide the answer but also enhance your understanding of quadratic equations and their solution methods.
Problem Statement
We are tasked with finding all values of x that satisfy the quadratic equation:
5(x+5)² - 38 = -18
This equation involves a squared term, which is the hallmark of quadratic equations. To solve it, we will simplify the equation, isolate the squared term, and then use the square root property to find the solutions. This process involves careful algebraic manipulation and attention to detail to ensure accuracy. The following sections will detail each step of the solution.
Step-by-Step Solution
1. Simplify the Equation
Our initial goal is to simplify the equation by isolating the squared term. To do this, we begin by adding 38 to both sides of the equation. This operation maintains the equality and helps move us closer to isolating the term containing x.
5(x+5)² - 38 + 38 = -18 + 38
This simplifies to:
5(x+5)² = 20
This step is crucial because it eliminates the constant term on the left side, allowing us to focus on the squared term. The next step will involve further isolating the squared term by dividing both sides by the coefficient.
2. Isolate the Squared Term
To further isolate the squared term, we divide both sides of the equation by 5:
(5(x+5)²)/5 = 20/5
This simplifies to:
(x+5)² = 4
Now, we have the squared term completely isolated on the left side of the equation. This is a significant milestone in solving the equation, as we can now apply the square root property to find the values of x that satisfy the equation. The next step involves taking the square root of both sides.
3. Apply the Square Root Property
To eliminate the square, we take the square root of both sides of the equation. It's crucial to remember that taking the square root introduces two possible solutions: a positive and a negative root.
√((x+5)²) = ±√4
This simplifies to:
x + 5 = ±2
We now have two separate equations to solve for x: one for the positive square root and one for the negative square root. This is a critical step in finding all possible solutions for the quadratic equation.
4. Solve for x
We now have two linear equations to solve:
- x + 5 = 2
- x + 5 = -2
Let's solve each equation separately.
For the first equation, x + 5 = 2, we subtract 5 from both sides:
x + 5 - 5 = 2 - 5
This gives us:
x = -3
For the second equation, x + 5 = -2, we subtract 5 from both sides:
x + 5 - 5 = -2 - 5
This gives us:
x = -7
Thus, we have found two solutions for x: -3 and -7. These are the values that satisfy the original quadratic equation.
Final Answer
The solutions to the quadratic equation 5(x+5)² - 38 = -18 are:
x = -3 and x = -7
These values can be verified by substituting them back into the original equation to ensure they hold true. The solutions represent the points where the parabola represented by the quadratic equation intersects the x-axis. This completes the solution process, providing a comprehensive answer to the given problem.
Verification
To ensure our solutions are correct, we substitute x = -3 and x = -7 back into the original equation 5(x+5)² - 38 = -18.
Verification for x = -3
5((-3)+5)² - 38 = 5(2)² - 38 = 5(4) - 38 = 20 - 38 = -18
The equation holds true for x = -3.
Verification for x = -7
5((-7)+5)² - 38 = 5(-2)² - 38 = 5(4) - 38 = 20 - 38 = -18
The equation also holds true for x = -7.
Both solutions satisfy the original equation, confirming the accuracy of our calculations. This verification step is crucial in ensuring the correctness of the solutions, providing confidence in the final answer.
Conclusion
In this article, we successfully solved the quadratic equation 5(x+5)² - 38 = -18 by following a systematic approach. We simplified the equation, isolated the squared term, applied the square root property, and solved for x. Our solutions, x = -3 and x = -7, were verified by substituting them back into the original equation. This exercise demonstrates the importance of algebraic manipulation and careful attention to detail in solving quadratic equations. Mastering these techniques is crucial for success in mathematics and related fields. The step-by-step solution provided here serves as a valuable resource for understanding and solving similar quadratic equations. The ability to solve such equations is a fundamental skill in algebra and has wide-ranging applications in various scientific and engineering disciplines. This comprehensive guide not only provides the solution but also enhances the reader's understanding of the underlying principles and techniques involved in solving quadratic equations.
