Linear Quadratic System

euclid2 · #1 $Old$ December 3rd, 2008, 06:46 PM

Determine the restrictions on the y-intercept so that $y=3x^2+6x-1$ intersects with a line with slope 2 in more then one place

skeeter · #2 $Old$ December 3rd, 2008, 07:24 PM

let k = y-intercept for the line

points of intersection can be found by setting the linear and quadratic equations equal to each other ...

$3x^2 + 6x - 1 = 2x + k$

set the equation equal to 0 ...

$3x^2 + 4x - (1+k) = 0$

if the line intersects the parabola twice, then the above quadratic must have two distinct real roots.

for the quadratic to have two distinct real roots, the discriminant, $b^2-4ac > 0$

$4^2 - 4(3)[-(1+k)] > 0$

solve the inequality for k.