Question

Solve for \(x\) and check. $$\sqrt{x}+5=9$$

Short answer

The solution is \(x = 16\), and the check confirms it is correct.

Step-by-step solution

Isolate the Square Root Term

Subtract 5 from both sides of the equation to isolate the square root term. The equation becomes \[\sqrt{x} + 5 - 5 = 9 - 5\] which simplifies to \[\sqrt{x} = 4\].

Square Both Sides

Square both sides of the equation to eliminate the square root. This gives us \[(\sqrt{x})^2 = 4^2\], which simplifies to \[x = 16\].

Check the Solution

Substitute the value of \(x\) back into the original equation to check if it is correct: \[\sqrt{16} + 5 = 9\]. Since \(\sqrt{16} = 4\), the equation simplifies to \[4 + 5 = 9\], which is true. So the solution \(x = 16\) is correct.

Key concepts

Isolating the Square Root

When solving square root equations, the initial step is to isolate the square root expression on one side of the equation. This means you have to move all other terms to the opposite side. Why do we do this? It's so we create a clearer path to rid the equation of the square root.

For example, consider the equation \[\sqrt{x}+5=9\]. Here, you will subtract 5 from both sides resulting in \[\sqrt{x} = 4\]. Isolating the square root sets up the stage for the next critical step, which involves squaring both sides of the equation to solve for the variable under the square root symbol.

Squaring Both Sides

Once the square root is isolated, the next crucial step is squaring both sides of the equation. By raising both sides to the power of two, the square root is effectively removed. This is because the square root and the square are opposite operations; they cancel each other out the same way addition and subtraction do.

For the isolated square root such as \[\sqrt{x} = 4\], squaring both sides means you'll compute \[ (\sqrt{x})^2 = 4^2 \], which simplifies to \[x = 16\]. This method directly reveals the value of the variable we are aiming to find. It's important to perform the same operation to both sides to maintain the equation's balance.

Checking Solutions in Equations

After finding a potential solution to the equation, the verification step is indispensable. Why? Because when we manipulate equations by squaring or taking roots, we can introduce solutions that don't actually satisfy the original equation. This is known as an 'extraneous solution'.

To check the solution, we substitute the obtained value back into the original equation to see if it results in a true statement. Using our example with \[x = 16\], we check by substituting 16 in place of \(x\) to get \[\sqrt{16} + 5 = 9\]. Since \[\sqrt{16} = 4\], our check simplifies to \[4 + 5 = 9\], confirming that \(x = 16\) is indeed a correct solution. This step is our assurance that the solution is valid within the context of the original problem.