Problem 31 - Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. See Example 4. $$ \sqrt{5 x}+10=8 $$

Step-by-Step Solution

Answer

No solution.

Step 1: Isolate the Square Root

Start by isolating the square root on one side of the equation. Subtract 10 from both sides: \( \sqrt{5x} = 8 - 10 \). This simplifies to \( \sqrt{5x} = -2 \).

Step 2: Examine the Square Root

Note that a square root cannot be negative because square roots are defined to be non-negative for real numbers. Since \( \sqrt{5x} = -2 \) is not possible, no real numbers satisfy the equation.

Step 3: Conclude No Solutions

Because the square root term resulted in a negative expression, there are no values of \( x \) that can satisfy the original equation.

Key Concepts

Isolating Square Roots

When solving equations involving square roots, the first critical step is isolating the square root on one side. This often involves manipulating the equation using simple arithmetic like addition or subtraction.
For example, let's look at an equation like \( \sqrt{a} + b = c \). To isolate the square root, you need to remove the additional terms from the side where the square root resides. This means you would subtract \( b \) from both sides to achieve \( \sqrt{a} = c - b \).
Once the square root is isolated, this sets the stage for further solving. Isolation is key because it simplifies the equation, making the square root easier to deal with. Remember to perform the same arithmetic operation on both sides to maintain the balance.

• Isolating helps in simplifying the problem.
• Use arithmetic operations like subtraction or addition.
• Isolation is a preparatory step prior to solving the square root.

Extraneous Solutions

Extraneous solutions are results that emerge from the process of solving an equation but do not satisfy the original equation. This can happen particularly when dealing with square roots and quadratic equations.
For example, while manipulating square roots through squaring both sides to eliminate the root, it's possible to introduce solutions that don't actually fit the initial problem. It’s crucial to test all solutions in the original equation to ensure they satisfy the equation fully.
Consider an instance where after squaring, you get \( x = 4 \) and \( x = -4 \). Upon testing, only \( x = 4 \) might satisfy the original condition if negatives are not acceptable.

• Extraneous solutions are additional, often incorrect, results.
• They can be introduced during manipulation like squaring.
• Always test solutions in the original equation.

Non-Negative Square Roots

Square roots by definition are non-negative when dealing with real numbers. The principal square root function \( \sqrt{x} \) is defined only for non-negative outputs, meaning negative results are not possible.
In equations like \( \sqrt{y} = -k \), where \( k > 0 \), this results in no solution because there is no real number for which the square root is negative. Recognizing this nature helps in quickly identifying when an equation has no valid solutions.

• Principal square roots are always non-negative.
• Equations resulting in negative square roots have no real solution.
• Understand non-negativity to assess the solvability of equations.