https://file.aiquickdraw.com/mathbot/file/1727293134549i4zq7b8p.jpg

Let's start by addressing the question provided in the image. The given expression is:

\[
2x^2 - 6x + 9
\]

### Part (a): Express in the form \( a(x - b)^2 + c \)

We are asked to express the quadratic \( 2x^2 - 6x + 9 \) in the form \( a(x - b)^2 + c \), where \( a \), \( b \), and \( c \) are integers or fractions.

This is done by completing the square. Let's go through the steps:

1. **Factor out the coefficient of \( x^2 \) from the first two terms**:
   \[
   2x^2 - 6x + 9 = 2(x^2 - 3x) + 9
   \]

2. **Complete the square**:
   To complete the square inside the parentheses, we need to take half of the coefficient of \( x \), square it, and add and subtract it. The coefficient of \( x \) is \( -3 \), so half of it is \( -\frac{3}{2} \), and squaring that gives \( \left(\frac{-3}{2}\right)^2 = \frac{9}{4} \).

   So, add and subtract \( \frac{9}{4} \) inside the parentheses:
   \[
   2\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) + 9
   \]
   Simplify:
   \[
   2\left(\left(x - \frac{3}{2}\right)^2 - \frac{9}{4}\right) + 9
   \]

3. **Simplify the expression**:
   Distribute the \( 2 \) and simplify:
   \[
   2\left(x - \frac{3}{2}\right)^2 - 2 \times \frac{9}{4} + 9
   \]
   \[
   2\left(x - \frac{3}{2}\right)^2 - \frac{18}{4} + 9
   \]
   \[
   2\left(x - \frac{3}{2}\right)^2 - 4.5 + 9
   \]
   \[
   2\left(x - \frac{3}{2}\right)^2 + 4.5
   \]

Thus, the quadratic \( 2x^2 - 6x + 9 \) in the form \( a(x - b)^2 + c \) is:

\[
a = 2, \quad b = \frac{3}{2}, \quad c = 4.5
\]

### Part (b): Solve \( 2x^2 - 6x + 9 = 12.5 \)

Using the expression from part (a), we can solve the equation:

\[
2\left(x - \frac{3}{2}\right)^2 + 4.5 = 12.5
\]

1. Subtract \( 4.5 \) from both sides:
   \[
   2\left(x - \frac{3}{2}\right)^2 = 12.5 - 4.5
   \]
   \[
   2\left(x - \frac{3}{2}\right)^2 = 8
   \]

2. Divide by \( 2 \):
   \[
   \left(x - \frac{3}{2}\right)^2 = 4
   \]

3. Take the square root of both sides:
   \[
   x - \frac{3}{2} = \pm 2
   \]

4. Solve for \( x \):
   \[
   x = \frac{3}{2} + 2 \quad \text{or} \quad x = \frac{3}{2} - 2
   \]
   \[
   x = \frac{7}{2} \quad \text{or} \quad x = -\frac{1}{2}
   \]

Thus, the solutions to the equation are:

\[
x = \frac{7}{2} \quad \text{or} \quad x = -\frac{1}{2}
\]

---

Do you have any questions or would you like more details? Here are some related questions to explore:

1. How does completing the square help in solving quadratic equations?
2. Can you always use the method of completing the square, and how does it compare to the quadratic formula?
3. What are some real-world applications of quadratic functions?
4. Can you derive the vertex form of a quadratic equation directly from its standard form?
5. How do the values of \( a \), \( b \), and \( c \) affect the graph of the quadratic function?

**Tip:** Completing the square is also a useful technique when dealing with optimization problems in calculus, especially when finding the maximum or minimum of quadratic functions.

2x^2 - 6x + 9 can be written in the form a(x - b)^2 + c, where a, b and c are integers or fractions in their simplest forms. 
(a) Work out the values of a, b and c. 
(b) Using your answer to part (a), solve 2x^2 - 6x + 9 = 12.5. 
Give any fractions or surds in your answers in their simplest forms.

Learn how to transform the quadratic expression 2x^2 - 6x + 9 into the form a(x - b)^2 + c using the completing the square method. This problem covers algebraic techniques essential for Grades 9-12 students and includes solving the equation 2x^2 - 6x + 9 = 12.5 with detailed steps.

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