91Ó°ÊÓ

To solve a radical equation, one of the first steps is to isolate the radical. A radical is a symbol like \( \sqrt{} \) (square root) that indicates the root is being taken. Isolating it means getting the radical by itself on one side of the equation.

In the equation \( \sqrt{x} + 8 = 0 \), the radical is \( \sqrt{x} \). To isolate it, you need to eliminate any additional terms on its side. You've got a \(+8\) that needs to be removed. This can be done by performing the inverse operation: subtracting 8 from both sides of the equation.

Here's how it works:

• Our goal is to leave only \( \sqrt{x} \) on the left.
• Subtract 8 from both sides: \( \sqrt{x} + 8 - 8 = 0 - 8 \).

Now you're left with \( \sqrt{x} = -8 \). This step is crucial because it allows us to solve for \( x \) through subsequent steps.

After isolating the radical, the next step is to remove the square root. This involves eliminating the square root symbol to solve for \( x \).

To remove the square root, we square both sides of the equation. Squaring is the opposite operation of taking a square root. Here's the process:

• If \( \sqrt{x} = -8 \), square both sides: \( (\sqrt{x})^2 = (-8)^2 \).
• This simplifies to \( x = 64 \) because the square of any number, positive or negative, results in a positive number.

It's important to realize that even though squaring both sides leads to a solution \( x = 64 \), you must remember that the original equation had a square root set equal to a negative number, which is not possible with real numbers.
Despite reaching \( x = 64 \), this solution will need verification, considering the context of square roots.

In solving radical equations, it's always essential to check the obtained solution. This ensures the solution satisfies the original equation and aligns with mathematical rules involving radicals.

For our problem, you have found \( x = 64 \). The next step is substituting this value back into the original equation \( \sqrt{x} + 8 = 0 \):

• Substitute \( x = 64 \) into the equation to get \( \sqrt{64} + 8 \).
• Calculate \( \sqrt{64} \), which equals 8.
• This gives \( 8 + 8 = 16 \), not 0.

This discrepancy indicates the solution does not satisfy the original equation, confirming that no real solution exists.
Remember, when isolating a square root, we must be mindful of the domain, as a root set to a negative value is outside the set of real numbers. It's vital to double-check solutions for radicals to validate their correctness.