Examples of no solution equations

If you have an equation of the form:

ax²+bx+c=0, and b²-4ac<0, no solutions exist.

Why?

ax²+bx=-c

x² + (b/a) * x=-c/a

(x+b / (2a)) ² - (b / (2a)) ²=-c/a

(x+b / (2a)) ²-b² / (4a²) = - c/a

(x+b / (2a)) ²=b² / (4a²) - c/a

(x+b / (2a)) ² = (b²-4ac) / (4a²)

x+b / (2a) = + or - (√ (b²-4ac)) / (2a)

b²-4ac is the discriminant of this quadratic equation, and as it sits underneath a square root sign, it cannot be equal to a value less than zero. If b²-4ac<0, no solutions exist.