Solving Equations without a Linear Term
Solving by Completing the Square
The roots of a quadratic equation are its solutions. A real root is a solution of an equation that is also a real number. Some quadratic equations do not have any real roots.One way to tell that an equation of the form has no real roots is by looking at the graph of the related quadratic function:
When a quadratic equation has no real roots, it has a pair of complex roots that are complex conjugates. One strategy for solving quadratic equations with complex roots is completing the square.
Completing the square is the process of adding a constant term to a quadratic expression to form a perfect square trinomial. A perfect square trinomial is a trinomial, or polynomial with three terms, that can be written as the square of a binomial. For example, is a perfect square trinomial because its factored form is .To complete the square for a quadratic expression of the form , add the value of:
Solving by Using the Quadratic Formula
- If the discriminant is positive, then the quadratic equation has two real roots.
- If the discriminant is zero, then the quadratic equation has one real root.
- If the discriminant is negative, then the quadratic equation has two complex roots, which come in conjugate pairs.