Use the discriminant to determine types of roots
In the year 10 program, we learnt that the discriminant is b^2 - 4ac. We also learnt that if the discriminant is greater than 0, there are 2 real solutions. If the discriminant is equal to 0, there is only one real solution (perfect square). If the discriminant is less than 0, there are no real solutions (unreal roots).
Example 1.
![Picture](/uploads/1/1/5/3/11531538/4032731.png)
For this equation, we have to find values for k such that there are only real solutions. Therefore the discriminant must be greater than 0. k^2 - 16 > 0. k^2 > 16. Therefore k has to be greater than 4 and smaller than -4.
Example 2.
![Picture](/uploads/1/1/5/3/11531538/9235217.png)
Find the value(s) of m so that the line y = mx + m is tangent to the parabola y = x^2. As the line has to be tangent to the parabola, we know that only one real root may exist. Therefore the discriminant of the line must equal 0.
Therefore m^2 - 0 = 0. So m = 0
Putting this in the equation, we get y = 0, which is the line that is tangent to the parabola y = x^2.
Therefore m^2 - 0 = 0. So m = 0
Putting this in the equation, we get y = 0, which is the line that is tangent to the parabola y = x^2.