Problem 15 Solve each equation. $$ 4-\s... [FREE SOLUTION]

Chapter 10: Problem 15

Solve each equation. $$ 4-\sqrt{x-2}=0 $$

Short Answer

Expert verified

x = 18

Step by step solution

Isolate the Square Root

To start, isolate the square root term on one side of the equation. Move the constant term (4) to the other side.\[4 - \sqrt{x-2} = 0 \sqrt{x-2} = 4\]

Square Both Sides

Square both sides of the equation to eliminate the square root.\[(\sqrt{x-2})^2 = 4^2 \ x-2 = 16\]

Solve for x

Add 2 to both sides of the equation to solve for $x$.\[x - 2 + 2 = 16 + 2 \ x = 18\]

Verify the Solution

Substitute $x = 18$ back into the original equation to verify the solution.\[4 - \sqrt{18-2} = 0 \4 - \sqrt{16} = 0 \4 - 4 = 0\]Since both sides of the equation are equal, $x = 18$ is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolation of Square Root Terms

The first step in solving square root equations is to isolate the square root term. This means you need to move all other terms to the opposite side of the equation. Consider the equation 4 - $\sqrt{x-2} = 0$. Here, we need $\sqrt{x-2}$ by itself on one side. To achieve this, subtract 4 from both sides to get $\sqrt{x-2} = 4$. This sets up the equation for the next critical step, which is eliminating the square root.

Squaring Both Sides

Once you have isolated the square root term, the next step is to square both sides of the equation. Squaring will eliminate the square root. Given $\sqrt{x-2} = 4$, square both sides to get $ (\sqrt{x-2})^2 = 4^2$. Applying the squaring operation results in $x-2 = 16$. This simplification is crucial as it transforms the equation into a more familiar algebraic form.

Solution Verification

After solving for the variable, it's essential to verify your solution. In our example, we found that $x = 18$. To verify, substitute $x = 18$ back into the original equation: $4 - \sqrt{18-2} = 0$. Calculating inside the square root gives $\sqrt{16} = 4$. Substituting this back, you get $4 - 4 = 0$. Since both sides are equal, the solution $x = 18$ is verified.

Basic Algebraic Operations

Understanding basic algebraic operations is fundamental when solving equations. These operations include addition, subtraction, multiplication, and division. For our problem, after squaring both sides, we had $x - 2 = 16$. To solve for $x$, add 2 to both sides: $x = 16 + 2$, leading to $x = 18$. These simple yet vital operations allow you to manipulate and solve equations effectively.

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91影视

Solve each equation. $$ 4-\sqrt{x-2}=0 $$

Short Answer

Step by step solution

Isolate the Square Root

Square Both Sides

Solve for x

Verify the Solution

Key Concepts

Isolation of Square Root Terms

Squaring Both Sides

Solution Verification

Basic Algebraic Operations

One App. One Place for Learning.

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