Problem 82 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ 4=\sqrt{x+8}-\sqrt{x}+2 $$

Step-by-Step Solution

Answer

The solution is \(x = 1\). No extraneous solutions.

Step 1: Rearrange the Equation

Start by isolating one of the square roots. We can do this by subtracting 2 from both sides of the equation to get: \[4 - 2 = \sqrt{x+8} - \sqrt{x}\] which simplifies to \[2 = \sqrt{x+8} - \sqrt{x}\].

Step 2: Add Square Roots

Isolate one of the radicals on one side by moving \(\sqrt{x}\) to the other side of the equation, yielding:\[\sqrt{x+8} = 2 + \sqrt{x}\].

Step 3: Square Both Sides

Square both sides of the equation to eliminate the square roots, remembering that squaring a binomial involves the expansion formula \((a + b)^2 = a^2 + 2ab + b^2\): \[ (\sqrt{x+8})^2 = (2 + \sqrt{x})^2 \] \[ x+8 = 4 + 4\sqrt{x} + x \].

Step 4: Isolate the Square Root Term

Simplify and isolate \(4\sqrt{x}\) by subtracting \(x\) from both sides: \[ 8 = 4 + 4\sqrt{x} \] then subtract 4 from both sides to get: \[ 4 = 4\sqrt{x} \].

Step 5: Solve for Square Root

Divide both sides by 4 to solve for \(\sqrt{x}\): \[ \sqrt{x} = 1 \].

Step 6: Square the Result

Square both sides again to find \(x\): \[ (\sqrt{x})^2 = 1^2 \] simplifies to \[ x = 1 \].

Step 7: Check for Extraneous Solutions

Substitute \(x = 1\) back into the original equation to verify: \[ 4 = \sqrt{1 + 8} - \sqrt{1} + 2 \] simplifies to \[ 4 = 3 - 1 + 2 \] which is \[ 4 = 4 \]. Since the original equation is satisfied, \(x = 1\) is not extraneous.

Key Concepts

Solving Radical Equations

When dealing with radical equations, the main goal is to eliminate the radical (the square root in this case) so that the equation turns into a simpler, more straightforward form. This often involves re-arranging the terms and making use of squaring both sides of the equation. Let's break it down. - **Rearrange the Equation:** Start by isolating the radical. For this equation, subtract any constants from both sides to focus on the term with square roots.- **Isolate a Radical:** Move one of the square roots entirely to one side. This not only simplifies our equation further but sets it up nicely for the next step. - **Square Both Sides:** When you square, the square root sign disappears, freeing the term from under the root. This step is crucial but must be handled with care as it might introduce extra solutions. For example, starting from \(\sqrt{x+8} = 2 + \sqrt{x}\), squaring both sides results in an equation without radicals. Just remember, after squaring, the new equation can potentially have solutions that don't satisfy the original equation, which we'll check later.

Extraneous Solutions

Extraneous solutions are possible outcomes of solving radical equations that arise because of the squaring process. They are solutions that, while mathematically valid in the new form of the equation, do not satisfy the original equation. Here's a glance at what happens:- **Origin of Extraneous Solutions:** When you square both sides, you might introduce values that solve the squared equation but not the original.- **Verification Step:** Always substitute the solutions back into the original equation. This ensures they indeed work. For instance, after arriving at \(x = 1\) in our squared equation, substituting back into the original equation confirmed it worked. If a result doesn't fit, it's crossed out as extraneous. This is important to avoid assuming all mathematical solutions to the transformed equations are valid in the initial setup.

Square Roots in Equations

Square roots can make equations look tricky at first glance, but with a methodical approach, they become much simpler to handle. Here's how square roots play a role in equations like these:- **Isolation is Key:** Often, you need to isolate a square root to one side of an equation. This prepares for the eventual removal by squaring.- **Simplifying with Roots:** If there are multiple roots, like in our problem, simplify step-by-step by dealing with one root at a time. The initial equation \(\sqrt{x+8} - \sqrt{x} + 2 = 4\) seemed complex but simplifying boiled it down effectively. Removing the roots by squaring systematically attacks the complexity. Understanding that sometimes just rearranging terms and tackling one radical or root step at a time can simplify seemingly difficult problems immensely. Always ensure your last answer makes sense by checking back with the original equation.