2. The curve has equation y = x ^ 2 - 7x + 15

a. Express y in the form of (x - a) ^ 2 + b where a and b are constants.

b. Hence find the x that produce minimum value of y.

Let's solve the problem step by step.

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

Given the equation:

\[
y = x^2 - 7x + 15
\]

To express this in the form \( (x - a)^2 + b \), we need to complete the square.

1. **Rewrite the quadratic part:**

   Start by focusing on the \( x^2 - 7x \) part. We can complete the square as follows:

   \[
   x^2 - 7x = \left(x^2 - 7x + \left(\frac{-7}{2}\right)^2\right) - \left(\frac{-7}{2}\right)^2
   \]

   Simplifying inside the square:

   \[
   x^2 - 7x = \left(x - \frac{7}{2}\right)^2 - \left(\frac{7}{2}\right)^2
   \]

2. **Substitute back into the original equation:**

   Now, substitute this back into the original equation for \( y \):

   \[
   y = \left(x - \frac{7}{2}\right)^2 - \left(\frac{7}{2}\right)^2 + 15
   \]

   Simplify the constants:

   \[
   y = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 15
   \]

   Convert 15 into a fraction with denominator 4 to simplify further:

   \[
   y = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + \frac{60}{4}
   \]

   Combine the constants:

   \[
   y = \left(x - \frac{7}{2}\right)^2 + \frac{11}{4}
   \]

   Therefore, the equation in the form \( (x - a)^2 + b \) is:

   \[
   y = \left(x - \frac{7}{2}\right)^2 + \frac{11}{4}
   \]

   Here, \( a = \frac{7}{2} \) and \( b = \frac{11}{4} \).

### Part (b): Find the value of \( x \) that produces the minimum value of \( y \)

In the expression \( y = (x - a)^2 + b \), the minimum value of \( y \) occurs when the square term \( (x - a)^2 \) is zero because squares are always non-negative.

So, set:

\[
(x - a)^2 = 0 \quad \Rightarrow \quad x - a = 0 \quad \Rightarrow \quad x = a
\]

Using the value of \( a \) from part (a):

\[
x = \frac{7}{2}
\]

Thus, the value of \( x \) that produces the minimum value of \( y \) is \( x = \frac{7}{2} \) or \( x = 3.5 \).

---

Would you like further details or have any other questions? 

Here are some related questions you might find helpful:

1. How do you determine the vertex of a quadratic function in general?
2. What is the geometric significance of completing the square?
3. Can every quadratic equation be expressed in the form \( (x - a)^2 + b \)?
4. How can you find the minimum or maximum value of a quadratic function without completing the square?
5. How does the discriminant relate to the vertex of a quadratic function?

**Tip:** When completing the square, always ensure the coefficient of \( x^2 \) is 1. If it’s not, factor it out before proceeding.

Learn how to transform the quadratic equation y = x^2 - 7x + 15 into vertex form (x - a)^2 + b and find the x value that minimizes y. Detailed step-by-step solution included. Suitable for Grades 9-10.

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