Question

Two skaters are practicing at the same time on the same rink. A coordinate grid is superimposed on the ice. One skater follows the path y=-4x+18​, while the other skater follows the curve y=-3^2+17x .Find all the points where they might collide if they are not careful.

Answer 1

First, I believe there was an error in your equation. It seems like you wanted to write "y=-3x^2 + 17x". With this correction, to find the points where the two skaters could possibly collide, we need to set the two expressions for y equal to each other and solve for x:

-4x + 18 = -3x^2 + 17x

Rearranging the terms, we obtain:

3x^2 + 21x - 18 = 0

This is a quadratic equation in the form ax^2 + bx + c = 0. We can solve for x using the quadratic formula:

x = [-b ± sqrt(b^2 - 4ac)] / (2a)

In this equation, a = 3, b = 21, and c = -18. So we have:

x = [-21 ± sqrt((21)^2 - 4*3*(-18))] / (2*3)
x = [-21 ± sqrt(441 + 216)] / 6
x = [-21 ± sqrt(657)] / 6
x = [-21 ± sqrt(3*7*31)] / 6
x = [-21 ± sqrt(3)* sqrt(7)* sqrt(31)] / 6

As the square root of 657 is not a rational number, we can leave the solutions in this form for generality:

x1 = [-21 + sqrt(3)* sqrt(7)* sqrt(31)] / 6
x2 = [-21 - sqrt(3)* sqrt(7)* sqrt(31)] / 6

Substitute these x-values into either the linear equation y=-4x+18 or the quadratic equation y=-3x^2+17x to find the corresponding y-values.

Therefore, the points of possible collision are:

([-21 + sqrt(3)* sqrt(7)* sqrt(31)] / 6 , y1) and
([-21 - sqrt(3)* sqrt(7)* sqrt(31)] / 6 , y2)

where y1 and y2 are the y-values obtained from substituting the x-values into one of the original equations.