Quadratic

Definition

A quadratic is a polynomial of degree 2, typically in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratics have a characteristic parabolic graph that opens upwards if $a > 0$ and downwards if $a < 0$.

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. Quadratic equations may have two real solutions, one real solution, or no real solutions depending on the discriminant ($b^2 - 4ac$).
3. When solving systems involving quadratics, substitution or elimination methods can be used to find points of intersection.
4. Factoring is another method to solve quadratics when they can be expressed as $(px + q)(rx + s) = 0$.
5. $y = ax^2 + bx + c$ represents a parabola with its vertex at $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$.

Review Questions

• What is the quadratic formula and how is it derived?
• How does the discriminant determine the number of solutions for a quadratic equation?
• Describe how you would solve a system involving a linear equation and a quadratic equation.

Related terms

Discriminant: The expression $b^2 - 4ac$ in the quadratic formula which determines the nature of the roots of the quadratic equation.

Vertex: The highest or lowest point on the graph of a parabola, calculated as $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$.

Parabola: The graph of a quadratic function, which is U-shaped and symmetric about its vertex.