Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 2 \sqrt{b}=8 $$

Step-by-Step Solution

Answer

The solution set is \( \{16\} \).

Step 1: Isolate the Square Root

Start by isolating the square root on one side of the equation. The current equation is: \[2 \sqrt{b} = 8\]To isolate \( \sqrt{b} \), divide both sides of the equation by 2:\[\sqrt{b} = \frac{8}{2} = 4\]

Step 2: Eliminate the Square Root

To eliminate the square root, square both sides of the equation:\[(\sqrt{b})^2 = 4^2\]This simplifies to:\[b = 16\]

Step 3: Verify the Solution

To verify the solution, substitute \(b = 16\) back into the original equation:\[2 \sqrt{16} = 8\]Calculate \(\sqrt{16}\):\[2 \times 4 = 8\]Both sides of the equation are equal, confirming that \(b = 16\) is correct.

Step 4: Write the Solution Set

Express the solution as a solution set. Since \(b = 16\) satisfies the original equation, the solution set is:\[\{16\}\]

Key Concepts

Square Roots

Square roots can be a complex concept at first, but with a little practice, they become more intuitive. If you see the expression \(\sqrt{b}\), it means "the number which, when multiplied by itself, gives \(b\)." For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).
When solving equations involving square roots, our goal is often to get rid of the square root so that we can directly solve for the variable.

• Begin by isolating the square root term.
• Once isolated, square both sides of the equation to remove the square root.

Keep in mind, when squaring both sides, ensure that you are working accurately, as this step is crucial for finding the correct solution.

Solution Set

In mathematics, a solution set is a collection of all possible values that satisfy a given equation or inequality. When solving equations, once you've determined the value(s) of the variable, these values form the solution set. For the equation \(2\sqrt{b} = 8\), after solving, we found \(b = 16\).
After confirming that \(b = 16\) satisfies the original equation, this becomes our solution set: \(\{16\}\).

• Write solution sets using curly braces.
• Verify the proposed solution to ensure it satisfies the original equation.

Solution sets are important because they clearly communicate the valid solutions to a problem.

Equation Verification

Verifying an equation is a critical step in problem-solving that ensures our solution is correct. The verification process involves substituting the proposed solution back into the original equation to check that it satisfies the equation.For example, in the equation \(2\sqrt{b} = 8\):

• After solving, we found \(b = 16\).
• Substitute \(b = 16\) back into the equation to verify: \(2 \times \sqrt{16} = 8\).

Calculating \(\sqrt{16} = 4\), we get \(2 \times 4 = 8\), which holds true. Equation verification confirms that our solution \(b = 16\) is indeed correct, ensuring the reliability of our answer.