Question

Consider the equation 5x^2-10x+c=0. What values of c result in the equation having a complex solutions? Represent your answer on the number line

Answer 1

Answer:

When we have a quadratic equation:

a*x^2 + b*x + c = 0

There is something called the determinant, and this is:

D = b^2 - 4*a*c

If D < 0, then the we will have complex solutions.

In our case, we have

5*x^2 - 10*x + c = 0

Then the determinant is:

D = (-10)^2 - 4*5*c = 100 - 4*5*c

And we want this to be smaller than zero, then let's find the value of c such that the determinant is exactly zero:

D = 0 = 100 - 4*5*c

    4*5*c = 100  

       20*c = 100

            c = 100/20 = 5

As c is multiplicating the negative term in the equation, if c increases, then we will have that D < 0.

This means that c must be larger than 5 if we want to have complex solutions,

c > 5.

I can not represent this in your number line, but this would be represented with a white dot in the five, that extends infinitely to the right, something like the image below: