Problem 33 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{5-x}+10=9 $$

Step-by-Step Solution

Answer

There are no solutions because the square root cannot be negative.

Step 1: Isolate the Radical

To solve the equation \( \sqrt{5-x} + 10 = 9 \), we first need to isolate the term with the radical. Subtract 10 from both sides to get: \[ \sqrt{5-x} = -1 \].

Step 2: Check for Solvability

Consider the equation \( \sqrt{5-x} = -1 \). The square root function cannot produce a negative result, as it only outputs non-negative values. Therefore, there are no real solutions to this equation.

Key Concepts

Extraneous Solutions

When solving radical equations, we often encounter the term 'extraneous solutions'. But what are they? Extraneous solutions are values that may arise while algebraically manipulating an equation but do not actually satisfy the original equation.

This can happen during various stages of solving, particularly when both sides of an equation are squared. Squaring can introduce solutions that weren't valid in the original context. Therefore, it's crucial to always substitute back the solutions into the original equation to verify their validity. Doing this ensures that the solution isn't just a byproduct of the algebraic manipulation.

• Always check potential solutions in the original equation.
• Be mindful of algebraic steps that might produce extraneous results.

Isolating the Radical

A crucial step in solving radical equations is isolating the radical term. Imagine you have an equation entangled with radicals and constants, like how a knot mixes loops and ends.

Start by systematically untangling the equation to isolate the radical, which means getting it by itself on one side of the equation. In our example, we deal with the equation \( \sqrt{5-x} + 10 = 9 \). To isolate the radical term \( \sqrt{5-x} \), subtract 10 from both sides, yielding \( \sqrt{5-x} = -1 \). Now, the radical stands alone on one side.

This simplification helps you clearly see and solve the equation. Remember, isolating the radical term is often the very first and critical step in solving such problems.

• Subtract or add constants to one side to reveal the radical.
• Clear the equation step by step until the radical term is isolated.

Non-Negative Square Root Values

Understanding that square roots only produce non-negative results is key in solving equations involving squared terms. The square root function, by definition, is designed to give the primary or positive root of any number.

As in our exercise, the equation \( \sqrt{5-x} = -1 \) may at first seem solvable. However, since square root values cannot be negative, it's immediately apparent that no real numbers fulfill this equation.

This principle is essential because recognizing when a solution is invalid can save you from unnecessary calculations.

• Square roots are non-negative: Remember that they represent only the principal roots.
• Identifying invalid scenarios early can prevent extensive computations on unsolvable steps.