Quadratic equation

Definition

A quadratic equation is a polynomial equation of the form $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants, and $a \neq 0$. It describes a parabola in the Cartesian plane.

5 Must Know Facts For Your Next Test

1. The solutions to a quadratic equation can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
2. The discriminant, given by $\Delta = b^2 - 4ac$, determines the nature of the roots. If $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is one real root; if $\Delta < 0$, there are two complex roots.
3. Quadratic equations can also be solved by factoring or completing the square, provided they are factorable or easily manipulated.
4. In trigonometry, solving quadratic equations often involves trigonometric identities such as converting from sine or cosine forms to simpler expressions.
5. Graphically, the vertex form of a quadratic equation is useful for identifying key features like the vertex and axis of symmetry.

Review Questions

• What is the quadratic formula used to solve a quadratic equation?
• How does the discriminant determine the nature of the roots of a quadratic equation?
• What methods can be used to solve a quadratic equation besides using the quadratic formula?

Related terms

Discriminant: A value calculated from a quadratic equation's coefficients ($\Delta = b^2 - 4ac$) that indicates the nature of its roots.

Vertex Form: A way to express a quadratic function as $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex.

Trigonometric Identities: Equations involving trigonometric functions that are true for all values within their domains, often used to simplify trigonometric expressions.