How do you determine algebraically the value of k such that linear function g(x) =-3+k does not intersect the parabola f(x) =5x2 +3x -4?

Answer 1

Given the parabola: f(x)=5x²+3x-4

and the line: g(x)=-3+k (which is a horizontal line because there is no x component)

we can determine a value for k that will not intersect the parabola by first finding minimum y value for the parabola.

To do so, first find x=-b/2a = -3/(2*5) = -3/10 and plug this value in for x to determine the minimum y coordinate. So, f(-0.3)=5*(0.3)²+3*(0.3)-4 = -4.45.

If the minimum y value is -4.45, then we know any value less than this will not intersect the parabola.

Therefore g(x) must be less than -4.45: -3+k < -4.45

Solving for k gives us: k< -1.45.

Answer 2

Using completion of the square:

f(x) = 5(x + 0.3)² - 89/20

Therefore

-3 + k < -89/20

k < -89 / 20 + 3

k < -89 / 20 + 60 /20

k < -29 / 20