Question

Based on the quadratic formula, explain the solution to a quadratic equation if the value of b² is smaller than the value of 4ac.

Answer 1

Well, the quadratic formula is
Let say a quadratic equation is 3x^2+10x-25=0.
b = 10, a = 3, c = -25.
b^2 = 10^2=100
4ac=4(3)(-25)=-300
x = 5/3, -5
As shown, b^2 is larger than 4ac.
If b^2 is smaller than 4ac, then the value will be irrational, or an imaginary number.

Why?
Because if a smaller is subtracted by a larger number in a radical, then the value will be irrational.
For instance, b= 5, 4ac = 10
\[\sqrt{5-10} = \sqrt{-5}\]