Solving $\sqrt{10-13x} = X-4$: A Step-by-Step Guide

Hey guys! Today, let's dive into solving a radical equation that might seem a bit tricky at first glance. We're going to break down the steps to solve $\sqrt{10-13x} = x-4$. It's a classic algebra problem, and by the end of this article, you'll be a pro at tackling these types of questions. So, let's get started!

Understanding Radical Equations

Before we jump into the solution, let's quickly recap what radical equations are and why they can be a bit more challenging than your regular equations. Radical equations are equations where the variable is stuck inside a radical, most commonly a square root. The main challenge with these equations is that squaring both sides (which we'll need to do to get rid of the square root) can sometimes introduce solutions that aren't actually valid. These sneaky solutions are called extraneous solutions. So, it's super important to check our answers at the end to make sure they work!

When dealing with radical equations, always remember the key is isolating the radical term. Once you isolate the radical, you can raise both sides of the equation to the appropriate power to eliminate the radical. For square roots, you square both sides; for cube roots, you cube both sides, and so on. This step is crucial in simplifying the equation into a more manageable form. However, as we mentioned earlier, this process can sometimes lead to extraneous solutions, which is why checking your answers is a mandatory step.

Another important concept to keep in mind is the domain of the radical expression. In our specific equation, $\sqrt{10-13x}$, the expression inside the square root, $10-13x$, must be greater than or equal to zero. This is because we can only take the square root of non-negative numbers in the realm of real numbers. Ignoring this constraint can lead to incorrect solutions. So, always consider the domain restrictions when solving radical equations to ensure you're working within valid boundaries.

Step-by-Step Solution for $\sqrt{10-13x} = x-4$

Okay, let's get down to business and solve this equation step by step. Make sure you have your pen and paper ready so you can follow along!

Step 1: Isolate the Radical

The first thing we need to do, as we discussed, is to isolate the radical term. Luckily, in our equation, $\sqrt{10-13x} = x-4$, the radical is already isolated on the left side. This is a great start! If there were any terms added or subtracted on the same side as the radical, we would need to move them first. But in this case, we can move straight to the next step.

Step 2: Square Both Sides

Now, to get rid of the square root, we're going to square both sides of the equation. This means we'll have $(\sqrt{10-13x})^2 = (x-4)^2$. Squaring the left side is straightforward: the square root and the square cancel each other out, leaving us with $10-13x$. The right side requires a bit more attention. We need to expand $(x-4)^2$, which means multiplying $(x-4)$ by itself. Remember, $(x-4)^2$ is not the same as $x^2 - 4^2$; we need to use the FOIL method (First, Outer, Inner, Last) or the binomial square formula to expand it correctly.

When we expand $(x-4)^2$, we get $(x-4)(x-4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16$. So, our equation now looks like this: $10-13x = x^2 - 8x + 16$. We've successfully eliminated the radical, but now we have a quadratic equation to solve. Don't worry; we've got this!

Step 3: Rearrange into a Quadratic Equation

To solve the quadratic equation, we need to rearrange it into the standard form, which is $ax^2 + bx + c = 0$. To do this, we'll move all the terms to one side of the equation. Let's subtract $10$ and add $13x$ to both sides of the equation $10-13x = x^2 - 8x + 16$. This gives us:

$0 = x^2 - 8x + 16 - 10 + 13x$

Now, we'll simplify by combining like terms: $0 = x^2 + 5x + 6$. Great! We now have a quadratic equation in standard form. The next step is to solve it.

Step 4: Solve the Quadratic Equation

There are a few ways we can solve a quadratic equation: factoring, using the quadratic formula, or completing the square. In this case, factoring looks like the easiest method. We need to find two numbers that multiply to $6$ and add up to $5$. Those numbers are $2$ and $3$. So, we can factor the quadratic as follows:

$0 = (x + 2)(x + 3)$

To find the solutions, we set each factor equal to zero:

$x + 2 = 0$ or $x + 3 = 0$

Solving for $x$ in each case gives us:

$x = -2$ or $x = -3$

So, we have two potential solutions: $x = -2$ and $x = -3$. But remember, we're not done yet! We need to check for extraneous solutions.

Step 5: Check for Extraneous Solutions

This is the most crucial step! We need to plug each of our potential solutions back into the original equation, $\sqrt{10-13x} = x-4$, to see if they actually work. Let's start with $x = -2$:

$\sqrt{10 - 13(-2)} = -2 - 4$ $\sqrt{10 + 26} = -6$ $\sqrt{36} = -6$ $6 = -6$

This is not true! So, $x = -2$ is an extraneous solution. It doesn't work in the original equation.

Now, let's check $x = -3$:

$\sqrt{10 - 13(-3)} = -3 - 4$ $\sqrt{10 + 39} = -7$ $\sqrt{49} = -7$ $7 = -7$

This is also not true! So, $x = -3$ is also an extraneous solution.

Final Answer: No Solution

Since both potential solutions turned out to be extraneous, the equation $\sqrt{10-13x} = x-4$ has no solution. It's important to recognize when this happens, and checking your solutions is the key to avoiding this common pitfall.

Therefore, the correct answer is C. no solution.

Key Takeaways

Let's wrap up with some key takeaways from solving this equation:

• Isolate the radical: This is always the first step in solving radical equations.
• Square both sides: This eliminates the square root but can introduce extraneous solutions.
• Rearrange into a quadratic: If squaring both sides results in a quadratic equation, rearrange it into standard form.
• Solve the quadratic: Use factoring, the quadratic formula, or completing the square to find potential solutions.
• Check for extraneous solutions: This is the most important step! Plug your potential solutions back into the original equation to make sure they work.

By following these steps, you'll be well-equipped to tackle any radical equation that comes your way. Remember, practice makes perfect, so keep working on these types of problems to build your skills and confidence.

Alright guys, that's all for this problem! Keep practicing, and I'll catch you in the next one. Happy solving!