Quadratic equation

Definition

A quadratic equation is a polynomial equation of the second degree, typically expressed in the standard form as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ represents the variable. These equations are characterized by their parabolic graphs, which can open upwards or downwards depending on the sign of the leading coefficient, $a$. Quadratic equations often appear in various applications, including physics, engineering, and economics, making them essential to understand for problem-solving in real-world scenarios.

5 Must Know Facts For Your Next Test

1. Quadratic equations can have zero, one, or two real solutions depending on the value of the discriminant.
2. The graph of a quadratic equation is a parabola that can either open upward or downward based on the sign of $a$ in the equation $ax^2 + bx + c = 0$.
3. The quadratic formula, $x = \frac{{-b \pm \sqrt{D}}}{{2a}}$, is used to find the roots of any quadratic equation when factoring is not possible.
4. Completing the square is a method used to convert a quadratic equation into vertex form, which makes it easier to analyze its properties.
5. Quadratic equations frequently model situations such as projectile motion and profit maximization in business contexts.

Review Questions

• How does the discriminant affect the solutions of a quadratic equation?
  • The discriminant, calculated as $D = b^2 - 4ac$, plays a crucial role in determining the nature of the roots of a quadratic equation. If $D > 0$, there are two distinct real solutions; if $D = 0$, there is exactly one real solution (also called a repeated root); and if $D < 0$, there are no real solutions but two complex solutions. Understanding how to analyze the discriminant helps in predicting how many solutions exist without having to solve the equation directly.
• In what ways can you apply quadratic equations to solve real-world problems?
  • Quadratic equations can be applied in various fields such as physics for modeling projectile motion, where the path of an object follows a parabolic trajectory. In business, they are used to determine maximum profit by finding the vertex of a profit function. Additionally, they can be utilized in engineering to optimize dimensions for structures. By setting up relationships and translating them into quadratic equations, you can effectively analyze and solve these practical problems.
• Evaluate how understanding quadratic equations can influence decision-making in economic models.
  • Understanding quadratic equations allows economists to analyze functions that represent various economic scenarios such as cost, revenue, and profit. For instance, when maximizing profit or minimizing costs, identifying the vertex of the related quadratic function provides critical information on optimal production levels. This knowledge leads to better decision-making regarding resource allocation and pricing strategies. Therefore, mastery of quadratic equations equips individuals with analytical tools essential for informed decision-making in economics.

Related terms

discriminant: A component of the quadratic formula given by $D = b^2 - 4ac$, which determines the nature of the roots of a quadratic equation.

roots:

The solutions to a quadratic equation where the graph intersects the x-axis; they can be real or complex numbers.

vertex:

The highest or lowest point on the graph of a quadratic function, found at the point $(h, k)$ in vertex form $y = a(x - h)^2 + k$.