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 b^{2} -4ac

is used to determined the nature of the roots of a quadratic equation. It is called the discriminant.

If discriminant      = 0  , then the roots are rational and equal
                          <0  (negative), then the equation has two distinct unreal roots
                          >0  (positive), then the equation has two real roots
                                 ↳ if it's a perfect square, then the roots are rational, otherwise, the roots are irrational
1 5 1
b^2 -4ac is the one inside the square root sign in the quadratic formula. That is what makes it essential in determining the nature of the roots.
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