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We are tasked with finding the inverse of the function \( y = x^2 - 18x \) and matching it to the given form \( y = \pm \sqrt{bx + c} + d \).

### Step 1: Completing the square
To express the given quadratic in a form that can be inverted, we start by completing the square.

The original function is:
\[
y = x^2 - 18x
\]

Complete the square:
\[
y = (x^2 - 18x) = (x - 9)^2 - 81
\]
Thus, the equation becomes:
\[
y = (x - 9)^2 - 81
\]

### Step 2: Inverting the function
Now, solve for \(x\) to find the inverse:
\[
y + 81 = (x - 9)^2
\]
Take the square root of both sides:
\[
\pm \sqrt{y + 81} = x - 9
\]
Solve for \(x\):
\[
x = 9 \pm \sqrt{y + 81}
\]

Thus, the inverse function is:
\[
x = 9 \pm \sqrt{y + 81}
\]

### Step 3: Compare with the form given
The inverse is now in the form \( x = \pm \sqrt{by + c} + d \). Comparing it to the standard form \( y = \pm \sqrt{bx + c} + d \), we can match the coefficients:

- \( b = 1 \)
- \( c = 81 \)
- \( d = 9 \)

Thus, the values are:
\[
b = 1, \quad c = 81, \quad d = 9
\]

Would you like further clarification on any step? Here are some related questions to think about:

1. How does completing the square help in solving quadratic equations?
2. Why do we take the square root on both sides when inverting the function?
3. What is the general procedure for finding the inverse of a quadratic function?
4. How do you determine the positive or negative branch of the square root when inverting?
5. Can you find the domain and range of the inverse function?

**Tip:** When completing the square, always remember to balance the equation by adding and subtracting the same value to maintain equality.

Use the inverse of the function y = x^2 - 18x to find the unknown values for the equation y = ±√(bx + c) + d.

Find the inverse of the quadratic function y = x^2 - 18x through the method of completing the square. Learn to transform it into the form y = ±√(bx + c) + d and identify the values of b, c, and d. This problem covers advanced algebra concepts, ideal for students in Grades 10-12.

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