Properties of quadratic equations

Connect coefficients and discriminants with the number, type, and relationships of roots.

Meaning

For real-coefficient quadratics, determines the number of real roots. The two roots have sum and product For a≠0, completing the square gives (2ax+b)²=D. Since a real square cannot be negative, the sign of D determines the number of real roots. Expanding a(x−α)(x−β) for the roots α and β and comparing coefficients gives their sum and product.

Example

For so there is one distinct real root, 3. Counting the repeated root twice gives sum 6 and product 9.A negative discriminant means no real roots. It does not mean no solutions when complex numbers are allowed.

Practice for this concept

Scope and references

Topics

Discriminants, number and type of roots, roots and coefficients

Learning goal

Connect coefficients and discriminants with the number, type, and relationships of roots.

Helpful background

Explore the relationships between quadratic roots and coefficients.