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The problem you're working on involves solving the quadratic equation \(x^2 - 6x = 72\) using the **trial and improvement** method, and the solution lies between 4 and 5. Let me guide you through the process step by step.

### Step-by-step Process:

1. **Rewrite the equation**: The equation can be rearranged as:
   \[
   x^2 - 6x - 72 = 0
   \]
   Our goal is to find the value of \(x\) that satisfies this equation to one decimal place using trial and improvement.

2. **Trial and Improvement**: We will start by testing values between 4 and 5 for \(x\).

#### Trial 1: \(x = 4.5\)
   Substituting \(x = 4.5\) into the equation:
   \[
   (4.5)^2 - 6(4.5) = 20.25 - 27 = -6.75
   \]
   Since \(-6.75\) is less than 0, \(x = 4.5\) is too small.

#### Trial 2: \(x = 4.8\)
   Substituting \(x = 4.8\) into the equation:
   \[
   (4.8)^2 - 6(4.8) = 23.04 - 28.8 = -5.76
   \]
   Still less than 0, so \(x = 4.8\) is too small.

#### Trial 3: \(x = 4.9\)
   Substituting \(x = 4.9\) into the equation:
   \[
   (4.9)^2 - 6(4.9) = 24.01 - 29.4 = -5.39
   \]
   Still negative, but getting closer.

#### Trial 4: \(x = 4.95\)
   Substituting \(x = 4.95\) into the equation:
   \[
   (4.95)^2 - 6(4.95) = 24.50 - 29.7 = -5.2
   \]
   We're still not at 0, but closer. At this point, you can try to refine the guess further.

You should keep repeating this method until you find a solution where the left-hand side is as close to 0 as possible, ensuring to give the solution correct to one decimal place.

Would you like more details on any of these steps?

### Related Questions:
1. What is the significance of trial and improvement in solving equations?
2. How can we refine guesses in trial and improvement?
3. Can we use other methods, like the quadratic formula, to solve this?
4. Why does the equation have a solution between 4 and 5?
5. How does rounding affect the accuracy of trial and improvement?

**Tip**: Always check if there is a faster method (e.g., factoring or using the quadratic formula) before committing to trial and improvement for solving quadratic equations.

The equation x^2 - 6x = 72 has a solution between 4 and 5. Use a trial and improvement method to find this solution. Give your answer correct to 1 decimal place.

Learn how to solve the quadratic equation x^2 - 6x = 72 using the trial and improvement method. This solution guides you step-by-step, providing the solution to one decimal place. Ideal for students in Grades 9-11 working on algebraic skills and quadratic equations.

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