Problem 87 - Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{x-5}-\sqrt{x+3}=4 $$

Step-by-Step Solution

Answer

No solutions; the original equation has no real numbers that satisfy it.

Step 1: Isolate one square root

Start by isolating one of the square root terms. Add \( \sqrt{x + 3} \) to both sides of the equation: \[ \sqrt{x - 5} = \sqrt{x + 3} + 4 \]

Step 2: Square both sides

Square both sides of the equation to eliminate the square root on the left-hand side. This gives: \[ (\sqrt{x - 5})^2 = (\sqrt{x + 3} + 4)^2 \] This simplifies to: \[ x - 5 = x + 3 + 8\sqrt{x + 3} + 16 \] Therefore, the equation becomes: \[ -5 = 3 + 8\sqrt{x + 3} + 16 \] Simplifying the right side, we have: \[ -8 = 8\sqrt{x + 3} \]

Step 3: Solve for the square root term

Divide both sides by 8 to isolate the square root: \[ \sqrt{x + 3} = -1 \]

Step 4: Identify the impossibility

Since the square root of a number cannot be negative, there are no solutions to the equation. Therefore, the initial equation has no valid values of \( x \) since both sides squared incorrectly suggests a possible negative solution.

Key Concepts

Understanding Extraneous Solutions

When dealing with radical equations, it's crucial to recognize the potential for extraneous solutions. Extraneous solutions are results that appear to solve the equation but do not satisfy the original equation once substituted back in.
This often happens when both sides of an equation are squared. The squaring process can introduce solutions that weren't valid in the original equation, mainly because squaring eliminates any negative sign.
To handle this, always substitute any potential solutions back into the original equation. Check to see if they make a true statement possible. If not, they are extraneous and should be discarded.

The Importance of Square Root Isolation

Isolating a square root in a radical equation simplifies the solving process significantly. Think of it as untangling a knot, where separating the roots allows easier manipulation of the equation.
To isolate a square root, perform operations that place the square root on one side of the equation by itself. In the exercise provided, the square root \( \sqrt{x - 5} \) was isolated first, resulting in \( \sqrt{x - 5} = \sqrt{x + 3} + 4 \).

• This isolation step makes it easier to eliminate the square root by squaring both sides in the subsequent steps.
• Remember, isolating the term is vital to avoid errors and to ensure that any further operations are correctly applied.

This straightforward step can often determine whether the rest of the solution process is smooth or fraught with potential missteps.

Mastering Radical Equations

Radical equations contain variables located beneath a radical, most frequently the square root. Solving them often involves transforming these equations into polynomial equations through appropriate operations.
In our context, the solution process includes isolating a square root, squaring both sides of the equation to eliminate the radical, and then solving for the variable.

• Once the radicals are removed, the resulting equation can often be manipulated as a standard linear or quadratic equation.
• It’s vital to verify all solutions, as the process of elimination can introduce unwarranted solutions, leading to potential extraneous results.

Understanding how to work through these steps will enable you to tackle various radical equations effectively, making this a valuable skill in algebra.