Problem 6 - Algebra A Combined Function

Question

Solve. $$ \sqrt{5 x}=-5 $$

Step-by-Step Solution

Answer

No solution exists because \( \sqrt{5x} \) cannot be negative.

Step 1: Understand the Equation

The equation at hand is \( \sqrt{5x} = -5 \). We need to find \( x \) such that this equation holds true.

Step 2: Review the Properties of Square Roots

Recall that the square root function \( \sqrt{} \) only returns non-negative values, meaning \( \sqrt{5x} \) cannot be negative. This implies that there is a contradiction in the equation because it equates a non-negative number to -5.

Step 3: Identify the Impossibility

Since no real number can be squared to yield a negative result, and since the equation requires the square root to equal -5, it is clear that no solution exists.

Key Concepts

Properties of Square Roots

Square roots are fundamental in mathematics and understanding their properties is essential when solving equations like \( \sqrt{5x} = -5 \). A key property of square roots is that they always produce non-negative outputs. This means:

• The square root of any positive number is positive.
• The square root of zero is zero.
• Simplifying further, square roots yield non-negative results, making it impossible for \( \sqrt{5x} \) to be negative.

Therefore, when an equation like \( \sqrt{5x} = -5 \) is given, the left side being a square root is inherently non-negative, establishing an immediate discrepancy in the equation.

No Solution in Real Numbers

When analyzing equations involving square roots, especially those resulting in negative values like \( \sqrt{5x} = -5 \), notice how crucial it is to acknowledge that no real number can satisfy such conditions. Here’s why:

• Since the square root function only outputs non-negative values, and \(-5\) is a negative number, it is impossible for the equation to hold true within the set of real numbers.
• This rule holds regardless of the value of \( 5x \), because even if \( 5x \) is a positive number, the result cannot be negative.

Thus, recognizing this mismatch helps quickly determine that there is indeed no possible solution in real numbers, sidestepping any further unnecessary calculations.

Algebraic Contradictions

In mathematics, an algebraic contradiction occurs when an equation presents an inherent logical conflict. Equations like \( \sqrt{5x} = -5 \) are prime examples. Let's examine why:

• Algebraic contradictions arise when the algebraic expression and its expected output are fundamentally irreconcilable under given properties, such as with square roots.
• In this specific case, the contradiction is between a mathematically impossible task: making a naturally non-negative square root equal a negative number.

Such contradictions serve as clear indicators that an equation may have no solution. In solving problems, identifying contradictions early can save time and help maintain accuracy in mathematical problem-solving.