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.