Solve each equation by completing the square
r² = –12r - 35

r = - 7, r = - 5

Step-by-step explanation:

Given

r² = - 12r - 35 ( add 12r to both sides )

r² + 12r = - 35

To complete the square

add ( half the coefficient of the r- term )² to both sides

r² + 2(6)r + 36 = - 35 + 36

(r + 6)² = 1 ← take the square root of both sides )

r + 6 = ± 1 ( subtract 6 from both sides )

r = - 6 ± 1, thus

r = - 6 - 1 = - 7 or r = - 6 + 1 = - 5

The distance is:  

Step-by-step explanation:

We re-write the equation of the line in the format:

Notice we divided the fraction of y by 2/2, and the fraction of z by 3/3.

In that equation, the director vector of the line is built with the denominators of the equation of the line, thus:

Then the parametric equations of the line along that vector and passing through the point (-2, 3, -4) are:

We plug them into the equation of the plane to get the intersection of that line and the plane, since that intersection is the image on the plane of the point (-2, 3, -4)  parallel to the given line:

Then we solve that equation for t, to get:

Then plugging that value of t into the parametric equations of the line we get the coordinates of the intersection:

Then to find the distance we just use the distance formula:

So we get: