Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each value of \(x\) is a solution of the equation. Values (a) \(x=-3\) (b) \(x=0\) (c) \(x=21\) (d) \(x=32\) Equation $$\frac{\sqrt{x+4}}{6}+3=4$$

Step-by-Step Solution

Answer

This problem requires substitution and evaluation for each given value of \(x\). Correct results on substitution and evaluation will determine the correct solution sets.

Step 1: Check if \(x=-3\) is a solution

Substitute \(x=-3\) into the equation \(\frac{\sqrt{x+4}}{6}+3=4\). A true statement would confirm -3 as a solution. An incorrect statement would suggest otherwise.

Step 2: Check if \(x=0\) is a solution

Substitute \(x=0\) into the equation \(\frac{\sqrt{x+4}}{6}+3=4\). If the equation holds true then \(x=0\) is a solution. If not, then it is not.

Step 3: Check if \(x=21\) is a solution

Substitute \(x=21\) into the equation \(\frac{\sqrt{x+4}}{6}+3=4\). A correct result will confirm \(x=21\) as a solution.

Step 4: Check if \(x=32\) is a solution

Substitute \(x=32\) into the equation \(\frac{\sqrt{x+4}}{6}+3=4\). Evaluating this equation will determine if \(x=32\) is a solution.

Key Concepts

Equation Solutions

Understanding how to find equation solutions is fundamental in algebra. An 'equation' is a statement that two expressions are equal, and a 'solution' to an equation is a number that, when substituted into the equation, makes it a true statement. For example, consider the equation \(\frac{\sqrt{x+4}}{6}+3=4\). To determine if a value is a solution, we substitute it into the equation and simplify.

If the equation remains balanced and true, the value is indeed a solution. This process was applied in our exercise, where values like \(x=-3\), \(x=0\), \(x=21\), and \(x=32\) were tested against the given radical equation. For each value, we checked if it satisfied the equation, hence determining whether or not it is a solution.

Radicals in Mathematics

Dealing with radicals in mathematics, especially square roots, is a skill that requires understanding the properties of exponents and roots. A radical can be recognized by the \(\sqrt{}\) symbol and indicates that we are looking for a number which, when multiplied by itself, will give the number under the radical.

In the context of our equation, \(\sqrt{x+4}\) represents the square root of \(x+4\). When we evaluate square roots, remember that only the principal (non-negative) root is considered unless the negative is specified. Radicals can complicate equations, but with practice, they become manageable.

Square Root Functions

Square root functions contain a variable within a radical and they represent a relationship between two quantities, where one is a square root of the other. The function \(f(x) = \sqrt{x}\) is the simplest form, mapping each non-negative input \(x\) to its non-negative square root.

In our exercise, the square root function is slightly more complex, with the form \(f(x) = \frac{\sqrt{x+4}}{6}\). To solve square root functions in equations, one approach is to isolate the square root term on one side of the equation and then square both sides to eliminate the square root. However, always check for extraneous solutions when using this method.

Substitution Method in Algebra

The substitution method is commonly used to solve algebraic equations. It involves replacing a variable with a given number or expression. When a specific value is substituted into an equation, we can evaluate whether the equation holds true, thus indicating if the substitution is indeed a solution.

In the provided exercise, substitution helps test each value of \(x\) to assess if it satisfies the radical equation. Since we're working with radicals, ensure that the substituted number does not result in taking the square root of a negative number, as this would not yield a real number solution (unless dealing with the complex number system).