Problem 427 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{d}-2=\sqrt{d-20}\)

Step-by-Step Solution

Answer

d = 36

Step 1: Isolate one of the square root terms

Add 2 to both sides of the equation to isolate one of the square root terms: ewline \ \ \ \ { \( \sqrt{d}-2+2=\sqrt{d-20}+2 \) \ \( \sqrt{d}=\sqrt{d-20}+2 \)

Step 2: Square both sides

Square both sides of the equation to eliminate the square roots:ewline \ \ \( (\sqrt{d})^2=(\sqrt{d-20}+2)^2 \) \ \( d=(\sqrt{d-20}+2)^2 \) \ \( d=d-20+4\sqrt{d-20}+4 \) \ \( d=d-20+4\sqrt{d-20}+4 \)

Step 3: Simplify the equation

Subtract d and simplify the remaining equation: ewline \(0=-20+4\sqrt{d-20}+4 \) \ \(0=-16+4\sqrt{d-20} \)

Step 4: Isolate the remaining square root term

Add 16 to both sides: ewline \(16=4\sqrt{d-20} \) \ Divide by 4: \( 4=\sqrt{d-20} \)

Step 5: Square both sides again

Square both sides again to solve for d: ewline \( 4^2=(\sqrt{d-20})^2 \) \ \(16=d-20 \) \ Add 20 to both sides: \(36=d \)

Step 6: Verify your solution

Substitute d back into the original equation to ensure it is correct: ewline \( \sqrt{36}-2=\sqrt{36-20} \) \ \(6-2=\sqrt{16} \) \ \( 4=4 \) \ Since both sides are equal, the solution is verified.

Key Concepts

Isolating Square Roots

To solve square root equations like \(\sqrt{d} - 2 = \sqrt{d - 20}\), the first step is to isolate one of the square root terms. This means we need to get one of the square roots by itself on one side of the equation.

We start by adding 2 to both sides of the equation: \(\sqrt{d} - 2 + 2 = \sqrt{d - 20} + 2\)
This simplifies to: \(\sqrt{d} = \sqrt{d - 20} + 2\)

By isolating \(\sqrt{d}\), we set up a simpler equation to work with. This method can make the subsequent steps easier because we can now focus on just one square root term at a time. Always remember to perform the same operation on both sides of the equation to keep it balanced.

Squaring Both Sides

After isolating a square root, the next step is to eliminate it by squaring both sides of the equation. In our example: \(\sqrt{d} = \sqrt{d - 20} + 2\), we square both sides:

\( (\sqrt{d})^2 = (\sqrt{d - 20} + 2)^2 \)
This results in:
\( d = (\sqrt{d - 20} + 2)^2 \)

Squaring both sides gives: \( d = d - 20 + 4\sqrt{d - 20} + 4 \)
The key idea here is to eliminate the radicals, which allows us to work with polynomial equations that are typically easier to solve.
Proceed by simplifying and rearranging the equation: \(0 = -16 + 4\sqrt{d - 20} \). This brings us closer to solving for the variable.

Verifying Solutions

The final crucial step is to verify the solutions we derived. This ensures they are correct and satisfy the original equation. Using our solution \(d = 36\), we substitute it back into the original equation: \(\sqrt{36} - 2 = \sqrt{36 - 20}\):

\(6 - 2 = \sqrt{16} \)
This simplifies to:
\(4 = 4 \)

Since both sides of the equation are equal, our solution is verified. Verification is important because squaring both sides can sometimes introduce extraneous solutions, which are not valid in the original equation.
Double-checking through verification ensures that our solution is both correct and complete.