Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

After an algebra lesson, the following was left on the blackboard. The graph of the function y equals x squared and  2012 lines parallel to the line y equals x, each of which intersects the parabola in two points.  The sum of the x-coordinates of the points of intersection of the lines in the parabola is which of the following?  Each of our lines looks like one of these for some value of k, because it's parallel to y equals x, and then it has some y-intercept, whatever it is.  So now we want to solve the equation x plus k equals x squared, because we want to find the points of intersection of  the line and the parabola. So then let's move everything over to the other side to get x squared minus x minus k equals 0, and  now what we want is the sum of the x-coordinates of the points of intersection. That means we want the sum of the roots of this equation right here.  Well, we know by Vietta's formula that the sum of the roots is  negative of this coefficient divided by the leading coefficient right here. So let's call them x1 plus x2, the two roots. Well, this sum is equal to minus  negative 1 over 1, which is 1. So that means each of these lines contributes 1 to the total sum. We have 2012 such lines, so our answer has to be 2012.

Video Summary

The algebra lesson involves finding the sum of the x-coordinates where 2012 lines, each parallel to \( y = x \), intersect the parabola \( y = x^2 \). Each line can be represented as \( y = x + k \), intersecting the parabola at two points. Solving for the intersection gives the quadratic equation \( x^2 - x - k = 0 \), where the sum of the roots (x-coordinates) is 1, according to Vieta's formulas. With 2012 lines, each contributing a sum of 1, the total sum of all x-coordinates is 2012.

Keywords

algebra

parabola

intersection

x-coordinates

Vieta's formulas