Problem 34 - Algebra 2

Question

Solve. Check for extraneous solutions. \(3 \sqrt{2 x}-3=9\)

Step-by-Step Solution

Answer

The solution to the equation \(3 \sqrt{2 x}-3=9\) is \(x = 8\).

Step 1: Isolate the square root

You should begin by getting the square root by itself on one side of the equation. You can do this by adding '3' to each side of the given equation. By doing so, the equation simplifies to \(3 \sqrt{2 x} = 12\).

Step 2: Cancel out the square root

Now, to cancel the square root, square both sides of the equation. But before that, divide both sides of the equation by '3' to further isolate the square root. That will transform the equation into: \(\sqrt{2 x} = 4\). Now when you square both sides, you get \(2x = 16\).

Step 3: Solve for 'x'

Next, solve for 'x' by dividing both sides of the equation by '2'. This simplifies down to \(x = 8\). This is the potential solution for 'x', but it must be checked for validation.

Step 4: Check for extraneous solution

Substitute 'x = 8' back into the original equation to verify the solution found. If the LHS (left-hand side) equals the RHS (right-hand side), then the solution is valid. \(3 \sqrt{2*8}-3=9\) simplifies to \(3 * 4 - 3 = 9\) and then to \(12 - 3 = 9\), confirming that \(x = 8\) is indeed the solution.

Key Concepts

Extraneous Solutions

When solving radical equations, especially those involving square roots, it's crucial to check for extraneous solutions. These are solutions that emerge during the solving process but do not satisfy the original equation. They can occur when we square both sides of an equation since this can sometimes introduce new solutions that weren't present before.

A straightforward way to check for extraneous solutions is to substitute the solution back into the original equation. If both sides of the equation evaluate to the same value, the solution is valid. In the example problem, we solved for \(x = 8\). By substituting back into the original equation \(3 \sqrt{2x} - 3 = 9\), it confirms by simplifying that the solution is valid since \(3 \times 4 - 3 = 9\). Always performing this check will help you verify that your solutions are correct and not introduced errors.

Square Roots

Square roots are used in many mathematical problems, and understanding them is vital when dealing with radical equations. A square root of a number \(a\), written as \(\sqrt{a}\), is a value which, when multiplied by itself, gives \(a\). For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).

When you have an equation with a square root, such as \(\sqrt{2x}\), it implies that you're looking for a value that can replace the variable to make the equation true when squared. Solving such equations usually involves isolating the square root term and then eliminating the square root by squaring both sides. Care must always be taken since squaring can introduce extraneous solutions that need to be checked as shown above.

Solving Equations

Solving equations is a core aspect of algebra, involving finding the value of a variable that makes the equation true. The process typically involves manipulating the equation to isolate the variable on one side. This often requires inverse operations such as addition, subtraction, multiplication, or division.

In equations with radicals, such as the square root, you'll start by isolating the radical expression. In our example, isolating \(\sqrt{2x}\) was done by performing operations on both sides until we had \(\sqrt{2x} = 4\). After this, by squaring both sides of the equation, we eliminated the square root to solve for \(x\). Finally, simplification delivered the solution \(x = 8\). Remember always to backtrack and plug your solutions into the original equation to ensure accuracy and check for extraneous solutions.