If one root of the equation $$x^2 + px + 12 = 0$$ is $$4$$, while the equation $$x^2 + px + q = 0$$ has equal roots, then the value of '$$q$$' is

Let the first quadratic equation be:

$$ x^2 + px + 12 = 0 $$

Since 4 is a root of this equation, it must satisfy it. Substitute $$ x = 4 $$ into the equation:

$$ (4)^2 + p(4) + 12 = 0 $$

$$ 16 + 4p + 12 = 0 $$

$$ 28 + 4p = 0 $$

Solve for $$ p $$ by isolating the variable:

$$ 4p = -28 $$

$$ p = -7 $$

Now consider the second quadratic equation:

$$ x^2 + px + q = 0 $$

Substitute the value of $$ p = -7 $$ into this second equation:

$$ x^2 - 7x + q = 0 $$

For a quadratic equation to have equal roots, its discriminant must be equal to 0. The discriminant formula is:

$$ D = b^2 - 4ac = 0 $$

Identify the coefficients from the equation where $$ a = 1 $$, $$ b = -7 $$, and $$ c = q $$:

$$ (-7)^2 - 4(1)(q) = 0 $$

$$ 49 - 4q = 0 $$

Solve for $$ q $$ by isolating the variable:

$$ 4q = 49 $$

$$ q = \frac{49}{4} $$

Final Answer:

The value of q is $$ \frac{49}{4} $$.