91Ó°ÊÓ

To solve radical equations, such as \(\sqrt{x+5}-8=0\), a fundamental step is to 'isolate the radical'. This means that we need to get the radical expression, \(\sqrt{x+5}\), by itself on one side of the equation. We accomplish this by performing operations that reverse whatever is being done to the radical. In our example, \(\sqrt{x+5}\) is being subtracted by 8. To isolate \(\sqrt{x+5}\), we add 8 to both sides of the equation. This gives us \(\sqrt{x+5} = 8\), with the radical isolated. Isolating the radical makes the subsequent steps clearer and sets the stage for us to use other algebraic methods to solve the equation.

It's important to remember that the goal is to have the radical on one side and all other terms on the other. This makes the equation easier to work with and brings us one step closer to finding the value of the variable within the radical.

Once the radical is isolated, the next step in solving the equation is 'squaring both sides'. Squaring both sides of our equation \(\sqrt{x+5} = 8\) effectively removes the square root and gives us an equation that is easier to solve. Squaring the isolated radical, \(\sqrt{x+5}\), and the number on the other side of the equals sign, 8, we get \(\left(\sqrt{x+5}\right)^2 = 8^2\), which simplifies to \(x+5 = 64\).

When squaring both sides of the equation, it's crucial to apply the operation to the entire side correctly. Squaring both sides undoes the radical and turns our expression into a simpler algebraic equation. However, we must be cautious as squaring can introduce extraneous solutions, which are results that do not satisfy the original equation. Therefore, it is recommended to always check your solutions by substituting them back into the original equation.

With the radical gone, we are now left with a basic algebraic equation that can be solved through 'algebraic manipulation'. In our case, the new equation is \(x+5 = 64\), and it requires further simplification to solve for \(x\). We perform algebraic manipulation by doing the opposite operation to both sides of the equation to get \(x\). This involves subtracting 5 from both sides, resulting in \(x = 64 - 5\) which simplifies to \(x = 59\).

Algebraic manipulation is essentially a combination of addition, subtraction, multiplication, and division used to rearrange and simplify equations. It is the process by which we systematically apply these operations to isolate the variable and find its value. It's the cornerstone of solving algebraic equations, and grasping its concepts is essential for any student who wants to successfully navigate through more complex mathematical problems.