Problem 401 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{5 q+3}-4=0\)

Step-by-Step Solution

Answer

The solution to the equation is \(q = \frac{13}{5}\).

Step 1: Isolate the Square Root

Add 4 to both sides of the equation to isolate the square root term. \(\sqrt{5q + 3} - 4 + 4 = 0 + 4\) This simplifies to: \(\sqrt{5q + 3} = 4\)

Step 2: Eliminate the Square Root

Square both sides of the equation to eliminate the square root. \((\sqrt{5q + 3})^2 = 4^2\) This simplifies to: \(5q + 3 = 16\)

Step 3: Solve for q

Isolate the variable q. Subtract 3 from both sides of the equation: \(5q + 3 - 3 = 16 - 3\) This gives: \(5q = 13\) Next, divide both sides by 5: \(\frac{5q}{5} = \frac{13}{5}\) This simplifies to: \(q = \frac{13}{5}\)

Step 4: Verify the Solution

Substitute \(q = \frac{13}{5}\) back into the original equation to verify the solution: \(\sqrt{5\left(\frac{13}{5}\right) + 3} - 4 = 0\) This simplifies to: \(\sqrt{13 + 3} - 4 = 0\) Further simplifying: \(\sqrt{16} - 4 = 0\) Since \(\sqrt{16} = 4\), the equation holds true: \(4 - 4 = 0\)

Key Concepts

Isolating the Square Root

When solving radical equations, the first step is often to isolate the square root on one side of the equation. This means you want the square root term to be by itself. For example, in the equation \(\sqrt{5q + 3} - 4 = 0\), you need to get rid of the \(-4\) by adding 4 to both sides. This results in \(\sqrt{5q + 3} = 4\).

Isolating the square root is crucial because it prepares the equation for the next step: eliminating the square root. Without isolating it first, you would be unable to simplify the equation properly.

Eliminating the Square Root

Once the square root is isolated, the next goal is to eliminate it. This is done by squaring both sides of the equation. For instance, with the equation \(\sqrt{5q + 3} = 4\), if we square both sides, we get:

\[ (\sqrt{5q + 3})^2 = 4^2 \]

This simplifies to: \[ 5q + 3 = 16 \]

The squaring operation removes the square root, allowing you to solve the simplified algebraic equation. Always remember to simplify the equation fully after eliminating the square root.

Verifying the Solution

After you've solved for the variable, it's important to verify that your solution is correct. This step checks whether the solution works in the original equation. Substitution is the most effective manner to verify.

For example, if we solved and found \(q = \frac{13}{5}\), we substitute this value back into the original equation:

\sqrt{5 \left( \frac{13}{5} \right) + 3} - 4 = 0\

Breaking it down, we get: \sqrt{13 + 3} - 4 = 0\
\sqrt{16} - 4 = 0\, which simplifies to: \ 4 - 4 = 0\.

If both sides of the equation balance, then \(q = \frac{13}{5}\) is indeed the correct solution. Always verify because solutions can sometimes be extraneous when dealing with square roots.