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The given function is:

\[
g(x) = x^2 + 3px + (14p - 3)
\]

It is stated that the quadratic equation has two equal roots, meaning that the discriminant of the quadratic must be zero. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is given by:

\[
\Delta = b^2 - 4ac
\]

For the given quadratic:

- \(a = 1\)
- \(b = 3p\)
- \(c = 14p - 3\)

### Part (a): Finding the value of \(p\)

The discriminant must equal zero for the quadratic to have two equal roots:

\[
\Delta = (3p)^2 - 4(1)(14p - 3) = 0
\]

Simplifying this:

\[
9p^2 - 4(14p - 3) = 0
\]

\[
9p^2 - 56p + 12 = 0
\]

This is a quadratic equation in \(p\). Let's solve it using the quadratic formula:

\[
p = \frac{-(-56) \pm \sqrt{(-56)^2 - 4(9)(12)}}{2(9)}
\]

\[
p = \frac{56 \pm \sqrt{3136 - 432}}{18}
\]

\[
p = \frac{56 \pm \sqrt{2704}}{18}
\]

\[
p = \frac{56 \pm 52}{18}
\]

Now, solving for \(p\):

1. \(p = \frac{56 + 52}{18} = \frac{108}{18} = 6\)
2. \(p = \frac{56 - 52}{18} = \frac{4}{18} = \frac{2}{9}\)

Since \(p\) is stated to be an integer, the only valid solution is:

\[
p = 6
\]

### Part (b): Solving the quadratic equation for \(p = 6\)

Substitute \(p = 6\) into the equation \(x^2 + 3px + (14p - 3) = 0\):

\[
x^2 + 3(6)x + (14(6) - 3) = 0
\]

\[
x^2 + 18x + (84 - 3) = 0
\]

\[
x^2 + 18x + 81 = 0
\]

This simplifies to:

\[
(x + 9)^2 = 0
\]

Thus, the solution is:

\[
x = -9
\]

Let me know if you'd like further details or have any questions!

### Follow-up questions:
1. What are the implications of having two equal roots in a quadratic equation?
2. How is the discriminant used to determine the nature of the roots of a quadratic equation?
3. Can the quadratic formula be applied if the discriminant is negative? What does this imply about the roots?
4. What are the different forms of a quadratic equation, and how do they help in solving it?
5. How can you check if the solution \(x = -9\) is correct by substituting it back into the original equation?

**Tip:** Always check if an equation can be factored easily before applying the quadratic formula.

The function g(x) = x^2 + 3px + (14p - 3), where p is an integer, has two equal roots. a) Find the value of p. b) For this value of p, solve the equation x^2 + 3px + (14p - 3) = 0.

Explore how to solve the quadratic equation g(x) = x^2 + 3px + (14p - 3), where p is an integer, using the discriminant method. Learn how to find the value of p when the equation has two equal roots. Suitable for students in Grades 9-12.

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