Equal Roots Condition for the Equation

The given equation is $x(x + 2)(12-k) = 2$.

Expand the equation:

$(x^2 + 2x)(12-k) = 2$

$(12-k)x^2 + 2(12-k)x - 2 = 0$

This is a quadratic equation of the form $Ax^2 + Bx + C = 0$, where:

• $A = 12-k$
• $B = 2(12-k)$
• $C = -2$

For the equation to have equal roots, the discriminant ($\Delta$) must be zero.

$\Delta = B^2 - 4AC = 0$

Substitute the values of A, B, and C:

$(2(12-k))^2 - 4(12-k)(-2) = 0$

$4(12-k)^2 + 8(12-k) = 0$

Factor out $4(12-k)$:

$4(12-k)[(12-k) + 2] = 0$

$4(12-k)(14-k) = 0$

This implies either $12-k = 0$ or $14-k = 0$.

Therefore, $k=12$ or $k=14$.

If $k=12$, the original equation becomes $x(x+2)(0) = 2$, which simplifies to $0=2$. This is impossible.

Thus, we must have $k=14$.

Distance Calculation from Point to Line

The point is $(k, \frac{k}{2})$. Substituting $k=14$, the point is $(14, \frac{14}{2}) = (14, 7)$.

The line is given by $3x + 4y + 5 = 0$. Here, $A=3$, $B=4$, $C=5$.

The formula for the distance ($d$) from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is:

$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$

Substitute the coordinates of the point $(14, 7)$ and the line coefficients:

$d = \frac{|3(14) + 4(7) + 5|}{\sqrt{3^2 + 4^2}}$

$d = \frac{|42 + 28 + 5|}{\sqrt{9 + 16}}$

$d = \frac{|75|}{\sqrt{25}}$

$d = \frac{75}{5}$

$d = 15$

The distance is 15.