Conquer Radical Equations: A Step-by-Step Guide
Hey everyone, let's dive into the world of radical equations! These equations might seem a bit intimidating at first, but trust me, they're totally manageable with the right approach. Today, we're going to tackle the equation 2x - 5 = √(-26 + 9x). I'll break down each step so you can follow along easily and become a radical equation-solving pro. So, grab your pencils, and let's get started! This guide will walk you through the process, making sure you understand every step. We'll cover everything from squaring both sides to checking for extraneous solutions. Let's start by understanding the basics. A radical equation is simply an equation where the variable appears under a radical sign, such as a square root, cube root, or even higher roots. The key to solving these equations is to isolate the radical term and then eliminate the radical by raising both sides of the equation to the power that corresponds to the root. For instance, if we have a square root, we square both sides; if we have a cube root, we cube both sides, and so on. One of the common pitfalls in solving radical equations is the appearance of extraneous solutions. These are solutions that we obtain through our algebraic manipulations but don't actually satisfy the original equation. Therefore, it's super important to check your solutions at the end to ensure they are valid. Ready to become a radical equation solver? Let's get into it!
Step-by-Step Solution to 2x - 5 = √(-26 + 9x)
Alright, here's how we're going to conquer this equation. We'll methodically go through each step to find the solution. The key to mastering these types of equations is practice, practice, and more practice. With enough exercises, you'll start recognizing patterns and become very quick at solving them. Before we start, it's important to remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). We'll use this to make sure we don't make any mistakes when simplifying expressions. Now, let's get our hands dirty and start solving. Follow these steps and you'll get your answer.
Step 1: Isolate the Radical
In our equation, 2x - 5 = √(-26 + 9x), the radical term is already isolated on the right side of the equation. This makes our first step super easy – we don't have to do anything! Sometimes, you might have terms outside the radical that you need to move to the other side to isolate it, but not in this case. Think of it like a puzzle where the most difficult piece is already in place. This simplifies the whole process for us. Being able to quickly identify and isolate the radical is crucial. It means we can proceed to the next step without extra effort. Remember, the goal is to get the radical term all by itself on one side of the equation. That might mean adding or subtracting terms or even dividing by a coefficient. With enough practice, you will be an expert at this.
Step 2: Square Both Sides
Now that the radical is isolated, we can eliminate the square root by squaring both sides of the equation. So, we have:
(2x - 5)² = (√(-26 + 9x))²
When you square a square root, they essentially cancel each other out. On the left side, we need to expand (2x - 5)². Remember the formula: (a - b)² = a² - 2ab + b². Applying this, we get:
(2x)² - 2 * (2x) * 5 + 5² = -26 + 9x
This simplifies to:
4x² - 20x + 25 = -26 + 9x
Squaring both sides is a critical step, because it removes the radical. But, it can also introduce extraneous solutions, which is why the checking step is so important later on. Always remember to check your solutions in the original equation. Make sure you're meticulous in expanding and simplifying the expressions. Small mistakes can lead to the wrong answers. Don't rush this step. Take your time, write out each step clearly, and double-check your work.
Step 3: Simplify and Solve the Quadratic Equation
We now have a quadratic equation: 4x² - 20x + 25 = -26 + 9x. Let's simplify it by moving all terms to one side to set the equation to zero:
4x² - 20x - 9x + 25 + 26 = 0
Combining like terms, we get:
4x² - 29x + 51 = 0
Now, we need to solve this quadratic equation. We can try factoring, completing the square, or using the quadratic formula. In this case, let's try factoring. We're looking for two numbers that multiply to (4 * 51 = 204) and add up to -29. After some trial and error, we find that -12 and -17 satisfy these conditions. Therefore, we rewrite the middle term and factor by grouping:
4x² - 12x - 17x + 51 = 0
4x(x - 3) - 17(x - 3) = 0
(4x - 17)(x - 3) = 0
Now, we can set each factor equal to zero and solve for x:
4x - 17 = 0 => 4x = 17 => x = 17/4 x - 3 = 0 => x = 3
So, we have two potential solutions: x = 17/4 and x = 3. But, before we celebrate, we must check these solutions.
Step 4: Check for Extraneous Solutions
This is the most crucial step. We must plug our potential solutions back into the original equation: 2x - 5 = √(-26 + 9x), to see if they are valid. Let's start with x = 17/4:
2(17/4) - 5 = √(-26 + 9 * (17/4))
17/2 - 5 = √(-26 + 153/4)
(17 - 10)/2 = √( (-104 + 153)/4 )
7/2 = √(49/4)
7/2 = 7/2
This solution checks out! Now, let's check x = 3:
2(3) - 5 = √(-26 + 9 * 3)
6 - 5 = √(-26 + 27)
1 = √1
1 = 1
This solution also checks out! Both x = 17/4 and x = 3 are valid solutions to the original equation.
Final Answer
The solutions to the radical equation 2x - 5 = √(-26 + 9x) are x = 17/4 and x = 3. Congratulations! You've successfully solved a radical equation. Keep practicing, and you'll become a pro in no time. Remember to always check for extraneous solutions. This step is very important. Also, don't be afraid to break down complex equations into smaller, more manageable steps. Keep practicing, and you'll become very comfortable with solving radical equations.
