Problem 27 - Algebra A Combined Function

Question

Solve. $$ \sqrt{3 x+1}+2=0 $$

Step-by-Step Solution

Answer

There is no real solution to the equation.

Step 1: Isolate the Square Root

First, isolate the square root term on one side of the equation. To do that, subtract 2 from both sides of the equation: \[ \sqrt{3x + 1} + 2 = 0 \]Subtract 2 from both sides: \[\sqrt{3x + 1} = -2 \]

Step 2: Check for Real Solutions

The equation now is \( \sqrt{3x + 1} = -2 \). Recall that the square root function always yields a non-negative result (\( \geq 0 \)). Hence, the expression cannot equal -2, as there is no real number that when square rooted gives a negative result.

Key Concepts

Real Number Solutions

Before tackling an equation that contains a square root, it's essential to understand the concept of real number solutions. In mathematical terms, a real solution is a solution that can exist on the number line - the familiar plane stretching from negative to positive infinity without any breaks. Whether positive, negative, or zero, real solutions are touched on our daily experiences, making them very tangible. When solving square root equations, our goal is to find these real solutions. However, one has to be cautious — not all equations have real solutions. Particularly, equations involving square roots are subject to restrictions, as the square root function inherently requires its result to be a real and non-negative number. Hence, anytime you're solving such equations, always ponder whether treating a problem like a regular algebraic equation leads to reasonable results. This doesn't just make the solution valid but aligns with the foundational principles governing numbers and roots.

Isolation of Square Root

The first critical step in solving square root equations is to isolate the square root itself. If you think about the logic, it mirrors the process of peeling an onion — removing layers until you reach the core. Here’s how you can do it effectively:

• Start by isolating the square root by moving everything else to the opposite side of the equation. This may involve addition or subtraction.

In the given example, we have the equation: \[ \sqrt{3x + 1} + 2 = 0 \]By subtracting 2 from both sides, we attempt to isolate the square root:\[ \sqrt{3x + 1} = -2 \]At this point, we have isolated the square root term. However, upon isolation, it’s vital to assess the isolated term closely to confirm if the equation is still feasible to solve practically. Keeping the square root alone helps you focus and makes it much simpler to diagnose whether a solution can actually exist.

Non-negative Results

A fundamental principle when dealing with square roots is the understanding that these roots naturally aim to be non-negative. In simpler terms, the output of a square root function cannot be anything less than zero. This is because a square root essentially asks, "What number, when multiplied by itself, gives me this specific value?" Any number multiplied by itself must be positive, or zero, thereby making the square root of any real number non-negative.This property is a deal-breaker in our solving processes:

• Check if the isolated square root can equate with the number on the other side of the equation.
• If it’s not non-negative as per the requirement, then we realize there are no real solutions to the equation.

For our example, once the square root is isolated as:\[ \sqrt{3x + 1} = -2 \]We immediately spot a contradiction. The isolated value, \(-2\), is negative. Since a square root cannot equal a negative number, the conclusion is clear: there are no real number solutions to this problem. This understanding not only prevents wasting time but also builds strong foundational skills in recognizing possible versus impossible outcomes in mathematical solutions.