Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{a}=5 $$

Step-by-Step Solution

Answer

The solution set is \( \{ 25 \} \).

Step 1: Understand the Equation

The given equation is \( \sqrt{a} = 5 \). This equation means that when you take the square root of \( a \), you get 5. We need to find the value of \( a \) that satisfies this equation.

Step 2: Isolate the Variable

To isolate \( a \), we need to eliminate the square root by squaring both sides of the equation. This gives us:\[(\sqrt{a})^2 = 5^2\]This simplifies to:\[a = 25\]

Step 3: Check the Solution

We need to verify that our solution \( a = 25 \) is correct by substituting it back into the original equation:\[\sqrt{25} = 5\]Since \( \sqrt{25} = 5 \) is true, our solution is correct.

Step 4: Write the Solution Set

Since \( a = 25 \) satisfies the equation, the solution set is:\[\{ 25 \}\]

Key Concepts

Understanding the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. Understanding this concept is essential for solving equations involving square roots, such as the one given: \(\sqrt{a} = 5\). It simply means that some number \(a\) must be such that when you take its square root, you get 5.
To conceptualize this, think about the square root symbol \(\sqrt{}\) as a function. It's asking: "What number squared equals the number inside the symbol?" For example,

• The square root of 25 is 5 because \(5 \times 5 = 25\).
• The square root of 9 is 3 because \(3 \times 3 = 9\).

When we apply this to our problem, we find that \(\sqrt{a} = 5\) implies the need to determine a number whose square is 25.

Isolating the Variable

In algebra, isolating the variable means rearranging an equation so that the variable we want to solve for is by itself on one side of the equation. For our problem \(\sqrt{a} = 5\), we need to eliminate the square root to isolate \(a\).
The most common way to do this is by performing the inverse operation. The inverse of taking a square root is squaring both sides. Therefore, by squaring both sides of \(\sqrt{a} = 5\), you get:

• \((\sqrt{a})^2 = 5^2\)

Simplifying this equation yields \(a = 25\). Now, \(a\) is isolated, and we have solved for its value. This manipulation transforms a complex-looking equation into a simple one.

Writing the Solution Set

In mathematics, after solving an equation, it's important to present the results clearly. This is done by writing a solution set, especially in problems involving variables. A solution set contains all the possible values that the variable can take to satisfy the equation.
For the given problem \(\sqrt{a} = 5\), after finding \(a = 25\) as the solution, we state this in set notation. A solution set is usually written inside curly braces:

• \(\{a \mid a = 25\}\)

Or, for simplicity:

• \(\{25\}\)

This means that 25 is the only value that satisfies the given equation.

Checking Solutions

Checking solutions is an important final step in solving equations. It helps to verify the accuracy of the solution found by going back to the original equation. By substituting the solution back into the equation, you can ensure that both sides are equal, confirming correctness.
For the equation \(\sqrt{a} = 5\), having found \(a = 25\), we check by substituting 25 back into the original equation:

• \(\sqrt{25} = 5\)

Since the left side equals the right side, the solution is verified as correct. This step ensures that no mistakes were made during the solving process and instills confidence in the accuracy of the solution.