Question

find the solution to the equation, or explain why there is no solution. remember to check for extraneous solutions. (sqrt{4x - 1} = -2)

Explanation:

Step1: Recall square root property

The square root function \( \sqrt{y} \) (where \( y \) is the radicand) is defined such that the result is always non - negative for real numbers. That is, for any real number \( a \), \( \sqrt{a}\geq0 \) when \( a\geq0 \) (and undefined when \( a < 0 \)).
In the equation \( \sqrt{4x - 1}=-2 \), the left - hand side \( \sqrt{4x-1} \) represents a square root. By the property of square roots, \( \sqrt{4x - 1}\geq0 \) (provided that \( 4x-1\geq0 \), i.e., \( x\geq\frac{1}{4} \) for the square root to be defined in real numbers).

Step2: Analyze the right - hand side

The right - hand side of the equation is \( - 2 \), which is a negative number. Since a square root (of a real number) can never be negative, there is no real number \( x \) that can satisfy the equation \( \sqrt{4x - 1}=-2 \).

Answer:

There is no solution because the square root of a real number is always non - negative, and the right - hand side of the equation \( \sqrt{4x - 1}=-2 \) is negative.