Problem 4 - Algebra 2

Question

Solve. \(\sqrt{3 x+4}=4\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Isolate the square root

First, isolate the square root on one side of the equation. We already have \( \sqrt{3x + 4} = 4 \).

Step 2: Square both sides

Next, in order to remove the square root, square both sides of the equation. Doing this gives us: \( (\sqrt{3x + 4})^2 = 4^2 \), which simplifies to \(3x + 4 = 16\).

Step 3: Isolate x

Subtract 4 from both sides of the equations to keep it balanced, resulting in \(3x = 12\). Then, divide both sides by 3 to solve for x, giving \(x = 4 \).

Key Concepts

Isolating the Square Root

In solving radical equations, the first crucial step is isolating the square root. The idea is to have the square root expression on one side of the equation by itself. This makes it easier to address. For example, in the given equation \( \sqrt{3x + 4} = 4 \), the square root is already isolated, which simplifies the process significantly. While isolating the square root, be mindful of other terms that might initially be on the same side. If there were additional terms or coefficients, you would perform operations like addition, subtraction, multiplication, or division to move them to the other side of the equation. When working with more complex equations, you might need to manipulate the equation further. Always ensure the square root stands alone before proceeding to the next steps in the solution process.

Squaring Both Sides

Once the square root is isolated, the next step is to eliminate the radical by squaring both sides of the equation. Squaring is the opposite operation of taking the square root, and it allows us to remove the radical symbol. In our example, squaring both sides transforms the equation \((\sqrt{3x + 4})^2 = 4^2\). By doing so, we eliminate the square root, which simplifies to \(3x + 4 = 16\). Be careful when squaring. Remember that any changes made to one side of the equation must be applied to the other to maintain balance. This principle of equality is fundamental in algebra to ensure that the equation stays true.

Algebraic Manipulation

After removing the square root by squaring, algebraic manipulation comes into play to solve for the unknown variable. Here, you'll use various operations to isolate the variable on one side of the equation. From our example, after squaring, we have \(3x + 4 = 16\). First, you subtract 4 from both sides to get \(3x = 12\). This step ensures that x begins to stand alone.Finally, divide both sides by 3 to reach the solution, \(x = 4\). These steps involve simplifying and rearranging the equation logically. It's crucial to apply these operations systematically, always keeping the equation balanced. Remember, each action should move you closer to isolating the variable, giving you the solution straightforwardly.