when the solutions to each of the two equations below are graphed in the xy coordinate plane, the graphs of the solutions intersect at two places. Write the y-coordinate of the points of intersection in the boxes below in order from smallest to largest. y=2x and y=x^2-3
Accepted Solution
A:
The points of intersection are at (3, 6) and (-1, -2).
Since both of these equations have y isolated, we can set them equal to each other:
2x=x²-3
We want all of the variables on one side, so subtract 2x: 2x-2x = x²-3-2x 0=x²-3-2x
Write the quadratic in standard form: 0=x²-2x-3
This is easily factorable, as there are factors of -3 that will sum to -2. -3(1)=-3 and -3+1=-2: 0=(x-3)(x+1)
Using the zero product property we know that either x-3=0 or x+1=0; therefore x=3 or x=-1.
Substituting this into the first equation (it is simpler): y=2(3) = 6 y=2(-1)=-2
Therefore the coordinates are (3, 6) and (-1, -2).