Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation algebraically. Check your solutions. $$\sqrt{x-10}-4=0$$

Step-by-Step Solution

Answer

The solution for the given equation is \( x = 26 \).

Step 1: Organizing the equation

Start by making the square root term isolated on one side of the equation: \( \sqrt{x-10} = 4 \)

Step 2: Removing the square root

When you square both sides of the equation, you will get rid of the square root. It should give us:\( (\sqrt{x-10})^2 = 4^2 \)This simplifies to:\( x-10 = 16 \).

Step 3: Solving for x

To find the value of x, add 10 to both sides of the equation:\( x = 16+10 \)So you have:\( x = 26 \).

Step 4: Verify the solution

To check if \(x=26\) is a valid solution or not, substitute \(x = 26\) back into the original equation:\( \sqrt{26-10}-4=0 \)This simplifies to:\( \sqrt{16} - 4= 0 \)\( 4 - 4 = 0 \)The equation is true, so \(x=26\) is a valid solution for the original equation.

Key Concepts

Radical Equations

Radical equations involve variables under a radical sign, such as a square root or cube root. In our exercise, the equation \(\sqrt{x-10}-4=0\) includes a square root. Solving these equations requires careful manipulation to eliminate the radical.To solve a radical equation:

• First, isolate the radical expression. This ensures that all operations you perform affect only the radical term, making it easier to manipulate.
• Then, raise both sides of the equation to the power that corresponds with the radical. For a square root, you square both sides. This step is critical to eliminate the radical, leaving you with a more straightforward equation.

After these steps, you can proceed to solve the remaining equation as a linear equation.

Algebraic Manipulation

Algebraic manipulation is key to solving equations. It entails operations like addition, subtraction, multiplication, and division performed strategically to isolate variables.In the step-by-step solution:

• The equation \(\sqrt{x-10} = 4\) is squared to remove the square root, simplifying it to \(x-10=16\).
• Here, the trick is recognizing that squaring undoes the square root, turning an otherwise complex equation into a simple one.
• Next, you add 10 to each side to solve for \(x\). This action isolates \(x\), bringing us to our solution \(x=26\).

Remember, these algebraic techniques are about maintaining balance in the equation—whatever you do to one side, do to the other. This principle keeps the integrity of the equation intact.

Solution Verification

Solution verification is the process of validating that the solution derived is indeed correct. This step is crucial because operations like squaring can sometimes introduce extraneous solutions, which are solutions that do not actually satisfy the original equation.To verify the solution:

• Substitute the found solution back into the original equation. Replace \(x\) with 26 in \(\sqrt{x-10}-4=0\).
• Calculate \(\sqrt{26-10}\), which gives you \(\sqrt{16}\), equal to 4.
• Then, \(4-4\) simplifies to zero, confirming that \(x=26\) satisfies the equation, completing the verification process.

This simple plug-and-check method ensures your solution fits and is not an artifact of earlier manipulations.