Quadratic equation

Definition

A quadratic equation is a second-degree polynomial equation in a single variable, typically written as $ax^2 + bx + c = 0$, where $a \neq 0$. The solutions to the quadratic equation are known as the roots of the equation.

5 Must Know Facts For Your Next Test

1. Quadratic equations can be solved using factoring, completing the square, or the quadratic formula.
2. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
3. The discriminant ($b^2 - 4ac$) determines the nature and number of roots: if it is positive, there are two real roots; if it is zero, there is one real root; if it is negative, there are two complex roots.
4. The graph of a quadratic equation is a parabola that opens upwards if $a > 0$ and downwards if $a < 0$.
5. The vertex form of a quadratic equation is $y = a(x-h)^2 + k$, where $(h, k)$ represents the vertex of the parabola.

Review Questions

• What methods can be used to solve a quadratic equation?
• How does the discriminant affect the number and type of roots?
• What does the vertex form of a quadratic equation represent?

Related terms

Discriminant: In a quadratic equation $ax^2 + bx + c = 0$, the discriminant is given by $b^2 - 4ac$ and determines the nature of the roots.

Vertex: The highest or lowest point on the graph of a parabola represented by $(h, k)$ in its vertex form.

Parabola: The U-shaped curve that represents the graph of a quadratic function.