write in the form of (ax + b)^2 + c:
(i) 7+6x-9x^2


To write the expression \( 7 + 6x - 9x^2 \) in the form \( (ax + b)^2 + c \), we can follow these steps:

### Step 1: Rearrange the expression
First, rewrite the quadratic expression in a standard form:
\[
-9x^2 + 6x + 7
\]
The coefficient of \( x^2 \) is negative, so factor out the negative sign from the quadratic terms:

\[
-(9x^2 - \frac{6}{9}x) + 7
\]

### Step 2: Complete the square
We now need to complete the square for the expression inside the parentheses:
\[
9(x^2 - \frac{2}{3}x)
\]
To complete the square, take half of the coefficient of \( x \), square it, and add and subtract it inside the parentheses:
\[
9\left(x^2 - \frac{2}{3}x + \left(\frac{1}{3}\right)^2 - \left(\frac{1}{3}\right)^2\right)
\]
This simplifies to:
\[
9\left[\left(x - \frac{1}{3}\right)^2 - \left(\frac{1}{3}\right)^2\right]
\]
Which gives:
\[
9\left(x - \frac{1}{3}\right)^2 - 9\left(\frac{1}{3}\right)^2 = 9\left(x - \frac{1}{3}\right)^2 - 1
\]

### Step 3: Rewrite the expression
Now, substitute this back into the original expression:
\[
- \left[9\left(x - \frac{1}{3}\right)^2 - 1\right] + 7
\]
Simplify it:
\[
-9\left(x - \frac{1}{3}\right)^2 + 1 + 7
\]
Which finally gives:
\[
-9\left(x - \frac{1}{3}\right)^2 + 8
\]

So, the expression in the form \( (ax + b)^2 + c \) is:
\[
-9\left(x - \frac{1}{3}\right)^2 + 8
\]

Would you like any further details or clarification? Here are some related questions:

1. How do you complete the square for more complex quadratics?
2. Why do we factor out the negative sign in this case?
3. How do we convert a completed square back into its expanded form?
4. What happens when the quadratic coefficient isn't a perfect square?
5. How do you apply this method to solve quadratic equations?

**Tip:** Completing the square is especially useful for rewriting quadratic functions to find their vertex form, which helps identify maxima and minima easily.

write in the form of (ax + b)^2 + c: (i) 7+6x-9x^2

Learn how to rewrite the quadratic expression 7 + 6x - 9x^2 in the form (ax + b)^2 + c using the method of completing the square. This detailed step-by-step solution is ideal for students in Grades 9-11 and covers key concepts in algebra and quadratic equations.

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