Let's solve these quadratic functions step-by-step, using both completing the square and x-intercepts methods as required.

2-50. Quadratic functions

(a) $f(x) = x^2 + 6x + 15$

Method: Completing the Square

1. Start with the quadratic equation: $f(x) = x^2 + 6x + 15$
2. To complete the square, focus on $x^2 + 6x$. Take half of the coefficient of $x$ (which is 6), square it: $\left(\frac{6}{2}\right)^2 = 9$.
3. Add and subtract this 9 inside the equation: $f(x) = (x^2 + 6x + 9) + 15 - 9 = (x + 3)^2 + 6$
4. Now, the equation is in vertex form: $f(x) = (x + 3)^2 + 6$

Vertex: $(-3, 6)$

Axis of symmetry: $x = -3$

(b) $y = x^2 - 4x + 9$

Method: Completing the Square

1. Start with the quadratic equation: $y = x^2 - 4x + 9$
2. To complete the square, focus on $x^2 - 4x$. Take half of the coefficient of $x$ (which is -4), square it: $\left(\frac{-4}{2}\right)^2 = 4$.
3. Add and subtract this 4 inside the equation: $y = (x^2 - 4x + 4) + 9 - 4 = (x - 2)^2 + 5$
4. Now, the equation is in vertex form: $y = (x - 2)^2 + 5$

Vertex: $(2, 5)$

Axis of symmetry: $x = 2$

(c) $f(x) = x^2 - 8x$

Method: Averaging the x-intercepts

1. Factor the equation: $f(x) = x(x - 8)$ The x-intercepts are $x = 0$ and $x = 8$.
2. Find the midpoint of the intercepts to find the axis of symmetry: $\text{Axis of symmetry} = \frac{0 + 8}{2} = 4$
3. Substitute $x = 4$ into the original equation to find the y-coordinate of the vertex: $f(4) = 4^2 - 8(4) = 16 - 32 = -16$

Vertex: $(4, -16)$

Axis of symmetry: $x = 4$

(d) $y = x^2 + 7x - 2$

Method: Completing the Square

1. Start with the quadratic equation: $y = x^2 + 7x - 2$
2. To complete the square, focus on $x^2 + 7x$. Take half of the coefficient of $x$ (which is 7), square it: $\left(\frac{7}{2}\right)^2 = \frac{49}{4}$.
3. Add and subtract this $\frac{49}{4}$: $y = \left(x^2 + 7x + \frac{49}{4}\right) - 2 - \frac{49}{4}$ Simplify: $y = \left(x + \frac{7}{2}\right)^2 - \frac{57}{4}$
4. Now, the equation is in vertex form: $y = \left(x + \frac{7}{2}\right)^2 - \frac{57}{4}$

Vertex: $\left(-\frac{7}{2}, -\frac{57}{4}\right)$

Axis of symmetry: $x = -\frac{7}{2}$

2-51. Expression to complete the square

For an expression of the form $x^2 + bx$, the number you need to add to complete the square is $\left(\frac{b}{2}\right)^2$.

For example, if you have $x^2 + 6x$, you would add $\left(\frac{6}{2}\right)^2 = 9$ to complete the square.

Would you like more details on any of these steps? Let me know if you have any questions.

Here are 5 related questions to expand your understanding:

1. How can the method of completing the square be used to solve quadratic equations?
2. How does the vertex form of a quadratic function help in graphing parabolas?
3. What is the significance of the axis of symmetry in a quadratic function?
4. How does factoring a quadratic function relate to its x-intercepts?
5. Can completing the square be used for any quadratic equation, even those with non-integer coefficients?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value!