Problem 46 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt{x-4}+1=5$$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{x-4}+1=5\) is \(x=20\).

Step 1: Isolate the square root term

Subtract 1 from both sides of the equation to isolate the square root: \[\sqrt{x-4} = 4\]

Step 2: Square both sides of the equation

Square both sides of the equation to remove the square root: \[(\sqrt{x-4})^2 = 4^2\] Simplify: \[x - 4 = 16\]

Step 3: Solve the resulting linear equation

Add 4 to both sides to solve for x: \[x = 16 + 4\] \[x = 20\]

Step 4: Check the solution in the original equation

Substitute the solution back into the original equation to make sure it works: \[\sqrt{20-4}+1 = 5\] \[\sqrt{16}+1 = 5\] \[4+1=5\] \[5=5\] The solution is valid, so the solution to the equation \(\sqrt{x-4}+1=5\) is \(x=20\).

Key Concepts

Isolating the Square Root

When solving square root equations, the first step is to focus on isolating the square root term. This makes it easier to eliminate the square root in the following steps. In the given equation \( \sqrt{x-4} + 1 = 5 \), the square root is accompanied by a 1. By subtracting 1 from both sides of the equation, we isolate the square root on one side of the equation: \( \sqrt{x-4} = 4 \). Now, the square root term stands alone, making the equation much simpler and preparing us for the next step, which is to remove the square root by squaring both sides of the equation.

Squaring Both Sides

Once the square root term is isolated, the next step is to get rid of the square root by squaring both sides of the equation. This is an essential step because squaring a square root removes the radical, leaving you with the expression inside the root. In our equation, squaring both sides transforms \( \sqrt{x-4} = 4 \) into \( (\sqrt{x-4})^2 = 4^2 \).

• The left side simplifies to \( x-4 \), since the square and the square root cancel each other out.
• The right side simplifies to 16, since \( 4^2 = 16 \).

Now you have a much simpler equation: \( x - 4 = 16 \), which is a linear equation. This is why isolating the square root is an effective technique, as it clears the way for easier computation.

Linear Equations

With the square root eliminated, you're left with a linear equation to solve. Linear equations are much simpler because they involve variables to the first power. In our problem, once the square root has been squared away, the remaining equation is \( x - 4 = 16 \). Your goal is to solve for \( x \).

• To do this, add 4 to both sides of the equation to cancel out the \(-4\) next to \(x\).
• This gives you \( x = 16 + 4 \).
• Simplifying this, you end up with \( x = 20 \).

Now you have solved the linear equation, finding that \( x = 20 \). This is a straightforward process and demonstrates how simple the equation becomes once the square root is dealt with.

Checking the Solution

After solving the equation, it's important to verify that your solution is correct by substituting it back into the original equation. This ensures there were no errors made during calculations because sometimes, squaring both sides can introduce extraneous solutions.

• Start by replacing \( x \) with 20 in the original equation: \( \sqrt{20-4} + 1 = 5 \).
• Simplify inside the square root first: \( \sqrt{16} + 1 \).
• The square root of 16 is 4, so this becomes \( 4 + 1 \).
• Add them together to get 5, which equals the right side of the original equation.

Since both sides match, the solution \( x = 20 \) is indeed correct. Checking solutions is a good habit to have, especially in more complex calculations, to ensure that the answer is valid.