Problem 27 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Isolate the Square Root

Begin by isolating the square root term on one side of the equation. We have:\[ \sqrt{3x + 1} + 2 = 0 \]Subtract 2 from both sides to get:\[ \sqrt{3x + 1} = -2 \]

Step 2: Recognize the Impossibility

Recognize that the expression \( \sqrt{3x + 1} \) represents a square root, which is always non-negative for real numbers. Thus, it cannot be equal to \(-2\).

Step 3: Conclusion

Since there is no real value of \( x \) that will satisfy the equation \( \sqrt{3x + 1} = -2 \), the equation has no real solutions.

Key Concepts

Isolating Square Root

When solving square root equations, isolating the square root can make things easier. In any equation with a square root, start by getting the square root term alone on one side. This means you want the term with the square root to be by itself, without any other numbers or variables attached.

• Look at the starting equation: \( \sqrt{3x + 1} + 2 = 0 \).
• To isolate \( \sqrt{3x + 1} \), subtract 2 from both sides to achieve \( \sqrt{3x + 1} = -2 \).

Doing this helps you see exactly what the square root equals. It's a technique allowing easy analysis and understanding of what the next steps should be.

No Real Solutions

In math, when dealing with real numbers, a square root can never be negative. This is because the square root of any number is defined as the principal square root which is always non-negative.

• We isolated \( \sqrt{3x + 1} = -2 \).
• Notice, the right side is a negative number.
• It is critical to remember that for real numbers, square roots are non-negative.

Since the result on the right, \( -2 \), is negative, it indicates that it is impossible to find a real number \( x \) satisfying the equation. Hence, there are no real solutions. This understanding prevents us from trying methods that only work on real numbers.

Mathematical Impossibility

The term 'mathematical impossibility' applies when a solution cannot exist following the rules of mathematics. In this equation, \( \sqrt{3x + 1} = -2 \), the result is impossible because no real number can satisfy it.

• Square roots reflect the distance from zero, which is why they are non-negative.
• Attempting to square both sides would traditionally yield the solution, but here it would still lead to impossible results with real numbers.

Recognizing this impossibility helps students avoid wasting time on methods that assume all values are real, reinforcing the importance of understanding the nature of numbers and possible solutions. When you're faced with a negative square root in the equation, you can confidently conclude that the equation has no solutions in the realm of real numbers.