Problem 44 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\sqrt{x}-\sqrt{x-12}=2$$

Step-by-Step Solution

Answer

x = 16

Step 1: Isolate one of the square root terms

Rewrite the equation to isolate one of the square roots. For this problem, isolate √x. \( \sqrt{x} = 2 + \sqrt{x-12} \)

Step 2: Square both sides of the equation

To eliminate the square roots, square both sides of the isolated equation. \( (\sqrt{x})^2 = (2 + \sqrt{x-12})^2 \) This simplifies to: \( x = 4 + 4\sqrt{x-12} + (x-12) \)

Step 3: Simplify and solve for x

Combine like terms and isolate the remaining square root. \( x = x - 8 + 4\sqrt{x-12} \) Subtract x from both sides: \( 0 = -8 + 4\sqrt{x-12} \) Add 8 to both sides: \( 8 = 4\sqrt{x-12} \)

Step 4: Solve the remaining equation

Divide both sides by 4 to isolate the square root term. \( \sqrt{x-12} = 2 \) Square both sides again to solve for x: \( (\sqrt{x-12})^2 = 2^2 \) \( x-12 = 4 \) Add 12 to both sides: \( x = 16 \)

Step 5: Verify the solution

Substitute x = 16 back into the original equation to ensure it holds true. \( \sqrt{16} - \sqrt{16-12} = 2 \) \( 4 - 2 = 2 \) Since both sides of the equation are equal, x = 16 is correct.

Key Concepts

Isolating Square Roots

When solving square root equations, it's important to start by isolating one of the square root terms. This allows us to deal with each square root separately.
In our example, the original equation is \( \sqrt{x} - \sqrt{x-12} = 2 \).
Our goal is to isolate \( \sqrt{x} \), and we do this by simply moving \( \sqrt{x-12} \) to the other side of the equation. This simplifies our equation to: \( \sqrt{x} = 2 + \sqrt{x-12} \).
Now that we have one square root by itself on one side of the equation, it becomes easier to handle mathematically.

Squaring Both Sides

verifyfilling steps problem.

Verifying Solutions

Last, important step to substitutions the roots. Verify ensures correctly computed
We check root(pls) be accurate, plug back original to ensure expectation remains consistent.
Example verification, pef substituted simplified officer simplified true< example equal.body:
\sqrt{16} - \sqrt{16-12} = 2 \) transformed root \( 4 - 2 =4 -2, correctly computed tremendous