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The expression is the discriminant, denoted by . Its sign determines the type of roots before the formula is evaluated completely.
If (or ), then the quadratic equation has two different real roots.
Example:
If (or ), then the quadratic equation has one real repeated root (two roots with the same value).
Example:
If (or ), then the quadratic equation has two distinct non-real complex roots. For real coefficients, they form a conjugate pair.
Here defines the imaginary unit.
Example:
If and are the roots of the quadratic equation , then:
These relationships can verify calculated roots or determine the coefficient ratios and . A specific equation is fixed once the leading coefficient is also known.
Classify and solve the equation . Then verify the roots using their sum and product.
The coefficients are , , and . Its discriminant is
Because , the equation has two non-real complex roots. Apply the quadratic formula and use :
The calculation gives:
Their sum and product agree with Vieta's formulas:
Both checks use the original coefficients, so they also catch a lost sign or an incorrect denominator in the quadratic formula.
Published: . Updated: .
