Solving $x - \sqrt{4-3x} = 0$ A Step-by-Step Guide

Hey guys! Today, we're diving into the world of solving equations, specifically focusing on the equation $x - \sqrt{4-3x} = 0$. This type of equation, which involves a square root, might seem tricky at first, but don't worry! We're going to break it down step by step, making sure you understand each part of the process. So, grab your pencils, and let's get started!

Understanding the Equation

First, let's take a good look at the equation we're dealing with: $x - \sqrt{4-3x} = 0$. The key thing to notice here is the square root. Square roots can sometimes introduce complications because they have restrictions on the values inside them. Specifically, the expression inside the square root (in this case, $4-3x$) must be greater than or equal to zero. This is because we can't take the square root of a negative number and get a real number result. Keeping this in mind from the beginning will help us avoid potential pitfalls later on.

So, before we even start manipulating the equation, let's figure out what values of $x$ are even allowed. We need to ensure that $4 - 3x \geq 0$. Let's solve this inequality:

$4 - 3x \geq 0$

Subtract 4 from both sides:

$-3x \geq -4$

Divide both sides by -3 (and remember to flip the inequality sign because we're dividing by a negative number):

$x \leq \frac{4}{3}$

This tells us that any solution we find for $x$ must be less than or equal to $\frac{4}{3}$. This is an important condition, and we'll come back to it later to check our solutions.

Isolating the Square Root

The next step in solving this equation is to isolate the square root term. This means we want to get the square root expression by itself on one side of the equation. In our case, we can do this by adding $\sqrt{4-3x}$ to both sides of the equation:

$x - \sqrt{4-3x} + \sqrt{4-3x} = 0 + \sqrt{4-3x}$

This simplifies to:

$x = \sqrt{4-3x}$

Now we have the square root term nicely isolated. This sets us up for the next step, which involves getting rid of the square root altogether.

Squaring Both Sides

To eliminate the square root, we'll square both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced. So, squaring both sides of $x = \sqrt{4-3x}$ gives us:

$(x)^2 = (\sqrt{4-3x})^2$

This simplifies to:

$x^2 = 4 - 3x$

Now we have a quadratic equation, which is something we know how to solve! This is a big step forward.

Solving the Quadratic Equation

To solve the quadratic equation $x^2 = 4 - 3x$, we first want to get everything on one side, setting the equation equal to zero. We can do this by adding $3x$ and subtracting 4 from both sides:

$x^2 + 3x - 4 = 0$

Now we have a standard quadratic equation in the form $ax^2 + bx + c = 0$. There are a few ways to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, factoring is the easiest approach.

We're looking for two numbers that multiply to -4 and add to 3. Those numbers are 4 and -1. So, we can factor the quadratic equation as follows:

$(x + 4)(x - 1) = 0$

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

$x + 4 = 0$ or $x - 1 = 0$

Solving these equations gives us two potential solutions:

$x = -4$ or $x = 1$

Checking for Extraneous Solutions

This is a crucial step when dealing with equations involving square roots. Squaring both sides can sometimes introduce solutions that don't actually work in the original equation. These are called extraneous solutions. We need to check each of our potential solutions in the original equation $x - \sqrt{4-3x} = 0$ to see if they are valid.

Let's start with $x = -4$:

$-4 - \sqrt{4 - 3(-4)} = 0$

$-4 - \sqrt{4 + 12} = 0$

$-4 - \sqrt{16} = 0$

$-4 - 4 = 0$

$-8 = 0$

This is not true, so $x = -4$ is an extraneous solution. We can discard it.

Now let's check $x = 1$:

$1 - \sqrt{4 - 3(1)} = 0$

$1 - \sqrt{4 - 3} = 0$

$1 - \sqrt{1} = 0$

$1 - 1 = 0$

$0 = 0$

This is true, so $x = 1$ is a valid solution.

Remember our earlier condition that $x \leq \frac{4}{3}$? Both of our potential solutions, -4 and 1, satisfy this condition. However, only $x=1$ satisfies the original equation.

The Final Solution

After all that work, we've found that the only solution to the equation $x - \sqrt{4-3x} = 0$ is $x = 1$. So, the solution set is {1}.

In summary, solving equations with square roots involves several key steps:

1. Identify the restrictions: Determine the values of x that make the expression inside the square root non-negative.
2. Isolate the square root: Get the square root term by itself on one side of the equation.
3. Square both sides: Eliminate the square root by squaring both sides of the equation.
4. Solve the resulting equation: This might be a linear equation, a quadratic equation, or another type of equation.
5. Check for extraneous solutions: Substitute each potential solution back into the original equation to make sure it works.

By following these steps carefully, you can confidently solve equations involving square roots. Keep practicing, and you'll become a pro in no time! Remember, math is a journey, not a destination. Enjoy the process!