91Ó°ÊÓ

To start solving any equation that contains a square root, our initial focus should be on isolating the square root term. The idea is to get the square root expression alone on one side of the equation.
In our example, we have: \(9 - \sqrt{4x + 1} = 0\)
The square root term here is \(\sqrt{4x + 1}\). Since it’s accompanied by a 9 on the left side, our objective is to remove this constant term. We can do this by moving the constant to the other side of the equation, which we achieve through basic arithmetic operations like addition or subtraction.
So, subtract 9 from both sides:
\[ 9 - \sqrt{4x + 1} - 9 = 0 - 9 \] Simplifying, we get: \(\sqrt{4x + 1} = 9\)
Now that \(\sqrt{4x + 1}\) is isolated, we’re ready to move on to the next step.

Once the square root term is isolated, the next logical step is to eliminate the square root. We accomplish this by squaring both sides of the equation.
In our example: \(\sqrt{4x + 1} = 9\)
To remove the square root, we square the left side and the right side of the equation, which gives us: \[ (\sqrt{4x + 1})^2 = 9^2 \]This simplifies to:
\[ 4x + 1 = 81 \] Notice that squaring the root cancels it out, leaving the expression inside the square root \(4x + 1\) fully intact.
On the right, we simply find the square of 9, which is 81. This step converts our square root equation into a standard linear equation, making it easier to solve.

After squaring both sides and converting the square root equation to a linear form, our task now is to solve for the unknown variable.
From the previously simplified equation: \[ 4x + 1 = 81 \] First, we need to isolate the term containing the variable x. We start by getting rid of the constant on the left side. Subtract 1 from both sides: \[ 4x + 1 - 1 = 81 - 1 \] Simplifying, we get: \[ 4x = 80 \]
To find the value of x, we divide both sides by the coefficient of x, which is 4: \[ x = \frac{80}{4} = 20 \]
By following these steps, we've successfully solved the equation \(9 - \sqrt{4x + 1} = 0\) and found that x equals 20.
Always remember to verify your solution by plugging it back into the original equation to ensure that both sides of the equation balance.