Let a and b be real numbers, where a<b, and let A= (a,a^2) and B=(b,b^2). The line AB (meaning the unique line that contains the point A and the point B) has x-intercept (-3/2,0) and y-intercept (0,3). Find a and b.

To find the values of a and b, we can find the equation of the line AB using the two given intercepts.

The equation of a line can be written in the form y = mx + c, where m is the slope and c is the y-intercept.

Given that the line AB has an x-intercept (-3/2, 0), we can substitute these values into the equation to find the slope.

0 = m * (-3/2) + c
0 = -3m/2 + c

Similarly, given that the line AB has a y-intercept (0, 3), we can substitute these values into the equation to find another equation involving m and c.

3 = m * 0 + c
3 = c

Now, we can substitute c = 3 into the equation derived from the x-intercept:

0 = -3m/2 + 3

To further simplify, we multiply both sides of the equation by 2:

0 = -3m + 6

Next, we isolate m:

3m = 6
m = 6/3
m = 2

Now that we know the slope, we can substitute it into one of the equations to find the value of c:

0 = -3(2) + c
0 = -6 + c
c = 6

Therefore, the equation of line AB is y = 2x + 6.

Given that A = (a, a^2) and B = (b, b^2), we can substitute these values into the equation y = 2x + 6 to find a and b:

For point A:
a^2 = 2a + 6
a^2 - 2a - 6 = 0

We can solve this quadratic equation to find the solutions for a.

For point B:
b^2 = 2b + 6
b^2 - 2b - 6 = 0

Similarly, we can solve this quadratic equation to find the solutions for b.