The given equation is:

\[
2x - 5 - 3\sqrt{2x - 5} + 2 = 0
\]

To solve for \( x \), let's follow these steps:

### Step 1: Simplify the equation
First, combine the constants:
\[
2x - 5 + 2 = 2x - 3
\]
So the equation becomes:
\[
2x - 3 - 3\sqrt{2x - 5} = 0
\]

### Step 2: Isolate the square root term
Move all terms without the square root to the other side:
\[
-3\sqrt{2x - 5} = - (2x - 3)
\]
Now divide both sides by -3:
\[
\sqrt{2x - 5} = \frac{2x - 3}{3}
\]

### Step 3: Square both sides to eliminate the square root
\[
2x - 5 = \left( \frac{2x - 3}{3} \right)^2
\]
Squaring the right side:
\[
2x - 5 = \frac{(2x - 3)^2}{9}
\]

### Step 4: Eliminate the fraction by multiplying both sides by 9
\[
9(2x - 5) = (2x - 3)^2
\]
Expanding both sides:
\[
18x - 45 = 4x^2 - 12x + 9
\]

### Step 5: Rearrange into a quadratic equation
Move all terms to one side:
\[
4x^2 - 12x + 9 - 18x + 45 = 0
\]
Simplifying:
\[
4x^2 - 30x + 54 = 0
\]

### Step 6: Solve the quadratic equation
First, divide the entire equation by 2:
\[
2x^2 - 15x + 27 = 0
\]

Now solve using the quadratic formula:
\[
x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(2)(27)}}{2(2)}
\]
\[
x = \frac{15 \pm \sqrt{225 - 216}}{4}
\]
\[
x = \frac{15 \pm \sqrt{9}}{4}
\]
\[
x = \frac{15 \pm 3}{4}
\]

### Step 7: Find the two possible solutions
\[
x_1 = \frac{15 + 3}{4} = \frac{18}{4} = 4.5
\]
\[
x_2 = \frac{15 - 3}{4} = \frac{12}{4} = 3
\]

### Step 8: Verify the solutions
We need to check both solutions in the original equation to ensure they work.

For \( x = 4.5 \):
\[
2(4.5) - 5 - 3\sqrt{2(4.5) - 5} + 2 = 9 - 5 - 3\sqrt{9 - 5} + 2 = 4 - 3\sqrt{4} + 2 = 4 - 6 + 2 = 0
\]
So, \( x = 4.5 \) is a valid solution.

For \( x = 3 \):
\[
2(3) - 5 - 3\sqrt{2(3) - 5} + 2 = 6 - 5 - 3\sqrt{6 - 5} + 2 = 1 - 3(1) + 2 = 0
\]
So, \( x = 3 \) is also a valid solution.

### Final Answer:
The solutions are \( x = 4.5 \) and \( x = 3 \).

Would you like more details on any step?

### Related Questions:
1. How do you apply the quadratic formula?
2. What are some common ways to solve equations involving square roots?
3. How do you simplify expressions with square roots?
4. How do you verify solutions to an equation?
5. What are the steps to factor a quadratic equation?

### Tip:
Always check all solutions in the original equation when dealing with square roots, as squaring can sometimes introduce extraneous solutions.

Learn how to solve the radical and quadratic equation 2x - 5 - 3√(2x - 5) + 2 = 0 with a detailed step-by-step breakdown. The solution uses concepts like solving radical equations and applying the quadratic formula, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation