Problem 21 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ \sqrt{x}-10=0 $$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{x}-10=0 \) is \( x=100 \).

Step 1: Isolate the square root function

We first need to isolate the square root function on one side of the equation. We do this by shifting '10' to the other side of the equation. So, we rewrite the equation \( \sqrt{x}-10=0 \) as \( \sqrt{x}=10 \)

Step 2: Square both sides of the equation

The square root function can be eliminated by squaring both sides of the equation. This is done because the opposite operation of square rooting is squaring. So, we square both sides of the resulting equation from Step 1 (\( \sqrt{x}=10 \)). This gives us \( (\sqrt{x})^2=10^2 \) which simplifies to \( x=100 \).

Step 3: Verify the solution

Finally, always validate your solution by substituting the result back into the original equation. So, replace 'x' in the original equation \( \sqrt{x}-10=0 \) with '100' and verify that the equation holds true. So, \( \sqrt{100} - 10 = 0 \), this simplifies to 10 - 10, which is indeed 0.

Key Concepts

Isolate the Square Root

When confronted with equations involving square roots, the first step is often to isolate the square root expression. To effectively isolate the square root, you must move all the other terms to the opposite side of the equation.

For example, in the equation \[ \sqrt{x} - 10 = 0 \], you would begin by adding 10 to both sides of the equation to get \[ \sqrt{x} = 10 \]. This is an important initial step because it simplifies the equation and gets it ready for the next phase, which involves removing the square root by squaring both sides. Ensuring that the square root stands alone makes this process much neater and less prone to errors.

Squaring Both Sides

Once you have isolated the square root on one side of the equation, the next step is to eliminate the square root by squaring both sides. Squaring is the reverse operation of taking a square root and will leave you with an equation that is usually easier to solve.

For instance, with \[ \sqrt{x} = 10 \], squaring both sides yields \[ (\sqrt{x})^2 = 10^2 \], simplifying to \[ x = 100 \]. This is the essence of squaring both sides; it allows us to find the value of the variable under the square root. Remember, when you square both sides, you must apply the squaring operation to the entire side of the equation and not just to the square root itself.

Verifying Solutions in Algebra

After solving square root equations, it's crucial to verify the solutions because some operations, like squaring both sides, can introduce additional solutions that don't satisfy the original equation.

To verify a solution, plug it back into the original equation and check if it holds true. Using the solution from our example, \[ x = 100 \], we substitute 'x' in the original equation \[ \sqrt{x} - 10 = 0 \] to get \[ \sqrt{100} - 10 = 0 \], which simplifies to \[ 10 - 10 = 0 \]. Since this is a true statement, \[ x = 100 \] is confirmed as a valid solution. This step is not only a good practice to confirm accuracy but also a valuable learning moment to reinforce understanding of how to solve square root equations.