General form

Definition

The general form of a quadratic function is expressed as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. It is used to identify the coefficients that determine the properties of the quadratic function.

5 Must Know Facts For Your Next Test

1. In the general form, $a$, $b$, and $c$ are called coefficients, with $a$ being the leading coefficient.
2. The discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots; if $\Delta > 0$, there are two distinct real roots.
3. A quadratic equation in general form can be converted to vertex form by completing the square.
4. The sum of the roots can be found using $-\frac{b}{a}$ and their product using $\frac{c}{a}$.
5. Setting up a quadratic equation in general form often helps in solving it using methods like factoring, completing the square, or applying the quadratic formula.

Review Questions

• What are the coefficients in the general form of a quadratic equation?
• How do you determine the nature of roots using the discriminant?
• Convert $2x^2 + 4x - 6 = 0$ into its vertex form.

Related terms

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

Discriminant: $\Delta = b^2 - 4ac$, used to determine how many and what type of roots a quadratic equation has.

Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$, used to find solutions for quadratic equations.