Solving (y+10)^2 - 24 = 0: A Step-by-Step Guide
Introduction
In this comprehensive guide, we'll walk you through the process of solving the equation (y+10)^2 - 24 = 0, where y represents a real number. This equation is a quadratic equation in disguise, and by using algebraic techniques, we can find the values of y that satisfy the equation. This detailed explanation aims to provide a clear understanding of each step involved, making it easy for anyone to follow along, regardless of their mathematical background. Whether you're a student tackling homework or simply brushing up on your algebra skills, this guide will provide you with the tools you need to solve similar problems with confidence. Understanding the fundamentals of solving quadratic equations is crucial in various fields, including engineering, physics, and computer science, making this a valuable skill to acquire.
Understanding the Equation
Before we dive into solving the equation, let's take a moment to understand its structure. The equation (y+10)^2 - 24 = 0 is a quadratic equation in vertex form. The term (y+10)^2 indicates a squared expression, which is a key characteristic of quadratic equations. The constant term -24 shifts the parabola vertically. To solve this equation, we need to isolate the variable y. This involves undoing the operations applied to y, starting from the outermost operation and working our way inwards. This systematic approach ensures that we maintain the equality and arrive at the correct solution. Recognizing the form of the equation is the first step towards choosing the appropriate solution method. In this case, we'll use the square root property to solve for y. This method is particularly efficient when the quadratic equation is in vertex form or can be easily manipulated into this form. By understanding the structure of the equation, we can anticipate the steps required to solve it and avoid common pitfalls. This foundational knowledge is essential for tackling more complex mathematical problems.
Step-by-Step Solution
Now, let's proceed with the step-by-step solution to the equation (y+10)^2 - 24 = 0. Each step is explained in detail to ensure clarity and understanding.
Step 1: Isolate the Squared Term
The first step in solving the equation is to isolate the squared term. This means getting the term (y+10)^2 by itself on one side of the equation. To do this, we need to eliminate the constant term -24. We can achieve this by adding 24 to both sides of the equation. This maintains the equality and moves us closer to isolating the variable y.
(y+10)^2 - 24 + 24 = 0 + 24
This simplifies to:
(y+10)^2 = 24
Step 2: Take the Square Root of Both Sides
Once we have isolated the squared term, the next step is to take the square root of both sides of the equation. This is a crucial step because it undoes the squaring operation and allows us to work with the expression inside the parentheses. Remember that when we take the square root of a number, we must consider both the positive and negative roots. This is because both the positive and negative square roots, when squared, will give the same positive number.
√(y+10)^2 = ±√24
This simplifies to:
y + 10 = ±√24
Step 3: Simplify the Square Root
Before we proceed further, let's simplify the square root. The square root of 24 (√24) can be simplified by finding the prime factorization of 24 and looking for perfect square factors. The prime factorization of 24 is 2 x 2 x 2 x 3, which can be written as 2^2 x 2 x 3. We can take the square root of 2^2, which is 2, and leave the remaining factors under the square root sign.
√24 = √(2^2 x 2 x 3) = 2√6
So, our equation now becomes:
y + 10 = ±2√6
Step 4: Isolate y
The final step in solving for y is to isolate the variable by subtracting 10 from both sides of the equation. This will give us the values of y that satisfy the original equation.
y + 10 - 10 = -10 ± 2√6
This gives us two solutions for y:
y = -10 + 2√6
y = -10 - 2√6
Final Solutions
Therefore, the final solutions for the equation (y+10)^2 - 24 = 0 are:
y = -10 + 2√6 y = -10 - 2√6
These are the exact solutions. If you need approximate decimal values, you can use a calculator to find the square root of 6 and perform the calculations.
Verification of Solutions
To ensure the accuracy of our solutions, it's always a good practice to verify them by substituting them back into the original equation. This step helps to catch any potential errors made during the solving process. Let's substitute each solution back into the equation (y+10)^2 - 24 = 0.
Verification of y = -10 + 2√6
Substitute y = -10 + 2√6 into the original equation:
((-10 + 2√6) + 10)^2 - 24 = 0
Simplify the expression:
(2√6)^2 - 24 = 0
(4 * 6) - 24 = 0
24 - 24 = 0
0 = 0
Since the equation holds true, the solution y = -10 + 2√6 is correct.
Verification of y = -10 - 2√6
Substitute y = -10 - 2√6 into the original equation:
((-10 - 2√6) + 10)^2 - 24 = 0
Simplify the expression:
(-2√6)^2 - 24 = 0
(4 * 6) - 24 = 0
24 - 24 = 0
0 = 0
Since the equation holds true, the solution y = -10 - 2√6 is also correct.
Alternative Methods for Solving Quadratic Equations
While we used the square root property to solve this specific equation, it's important to be aware of other methods for solving quadratic equations. The two most common alternative methods are factoring and using the quadratic formula. Each method has its advantages and disadvantages, and the best method to use often depends on the specific equation you're trying to solve.
Factoring
Factoring involves rewriting the quadratic equation in the form (ay + b)(cy + d) = 0. This method is effective when the quadratic expression can be easily factored. Once the equation is factored, you can set each factor equal to zero and solve for y. However, not all quadratic equations can be factored easily, and in some cases, factoring can be time-consuming or even impossible.
Quadratic Formula
The quadratic formula is a general formula that can be used to solve any quadratic equation in the form ay^2 + by + c = 0. The formula is:
y = (-b ± √(b^2 - 4ac)) / (2a)
This method is particularly useful when the equation cannot be factored easily or when you need to find decimal approximations of the solutions. The quadratic formula always works, but it can be more computationally intensive than other methods, especially for simpler equations.
Conclusion
In conclusion, we have successfully solved the equation (y+10)^2 - 24 = 0 for real number y. By isolating the squared term, taking the square root of both sides, simplifying the square root, and isolating y, we found the two solutions: y = -10 + 2√6 and y = -10 - 2√6. We also verified these solutions by substituting them back into the original equation. Understanding how to solve quadratic equations is a fundamental skill in mathematics, and this step-by-step guide provides a solid foundation for tackling similar problems. Remember to consider alternative methods such as factoring and the quadratic formula, and choose the method that best suits the specific equation you're solving. For further exploration and practice, you can check out resources like Khan Academy's Quadratic Equations section.
