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Chapter 10: Problem 23

Solve the given equation. $$\sqrt{3+x}+8=3$$

Short Answer

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Step by step solution

01

Isolate the square root term

The initial equation is \(\sqrt{3+x} + 8 = 3\). To isolate the square root term, subtract 8 from both sides of the equation:\[\sqrt{3+x} + 8 - 8 = 3 - 8\]This simplifies to:\[\sqrt{3+x} = -5\]
02

Recognize the impossibility in real numbers

The equation \(\sqrt{3+x} = -5\) suggests a negative value under a square root which is impossible for real numbers because the square root of any real number is always non-negative. Therefore, there are no real solutions to this equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isolating the square root
To solve equations involving square roots, the first step is to isolate the square root term. This usually means rearranging the equation so that the square root is by itself on one side.
For example, starting from the equation \( \sqrt{3+x} + 8 = 3 \), our goal is to get the square root term \( \sqrt{3+x} \) alone.
We do this by subtracting 8 from both sides: \[ \sqrt{3+x} + 8 - 8 = 3 - 8 \] Simplifying this gives: \[ \sqrt{3+x} = -5 \] Now we have isolated the square root term, but there is another important consideration here, which leads us to our next concept.
real numbers
In the realm of real numbers, the square root of any number is always non-negative. This means that when we see \( \sqrt{3+x} = -5 \), it signals an immediate issue.
Real numbers include all positive numbers, negative numbers, and zero. They do not include imaginary or complex numbers.
Therefore, \(-5\) cannot be the output of a square root operation on real numbers because the square root of a real number cannot be negative. This understanding tells us that there are no valid real solutions for the given equation as it stands.
solving algebraic equations
Solving algebraic equations involves finding the value of the variable that makes the equation true. Here, we start with the equation \( \sqrt{3+x} + 8 = 3 \).
Following the steps:
  • First, we isolated the square root term to obtain \( \sqrt{3+x} = -5 \).
  • Second, realizing that the equation \( \sqrt{3+x} = -5 \) can't have a solution in real numbers since the square root of a real number cannot be negative.
If you encounter similar equations, always remember to isolate the square root term first and then check if the resultant equality makes sense with real numbers. Some equations may have no real solutions, as in this case, and recognizing these situations is a crucial part of solving algebraic equations.

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Most popular questions from this chapter

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