Quadratic Equations and Functions Quiz

Quadratic Equations

Overview

A quadratic equation is a polynomial of degree 2, meaning it involves variables raised to the power of 2. Quadratic equations have the general form ax² + bx + c = 0, where a, b, and c are constants and x is the variable.

Solving Quadratic Equations

Solving a quadratic equation involves finding the values of x that make the equation true. There are several methods to solve quadratic equations, including factorization, completing the square, and using the quadratic formula.

Factorization

Factorization involves rearranging the polynomial into a product of two binomials. This method is most effective when the quadratic equation can be written in the form (x - p)(x - q) = 0, where p and q are constants.

Completing the Square

Completing the square involves adding and subtracting the square of half the coefficient of the quadratic term. The goal is to transform the equation into a perfect square trinomial.

Quadratic Formula

The quadratic formula is a general method to find the solutions of any quadratic equation. It is given by:

x = (-b ± √(b² - 4ac)) / 2a

This formula delivers two roots, one when the positive sign in front of the square root is chosen and the other when the negative sign is chosen.

Graphing Quadratic Functions

Quadratic functions are defined by the equation f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which is a continuous curve symmetric about the vertical axis. The parabola opens upwards if a is positive and opens downwards if a is negative.

Features of Quadratic Functions

Quadratic functions have the following features:

• Domain: All real numbers
• Range: All real numbers
• Vertex: (h, k), where h = -b / 2a and k = f(h)
• Symmetry: Symmetric with respect to the vertical axis
• Maximum or minimum point: Vertex

Quadratic Equations/Functions Word Problems

Quadratic equations and functions can be used to model real-world problems, such as the path of a thrown ball, the area of a circle, or the cost of producing a product. Solving these problems involves setting up the quadratic equation and using the appropriate method to find the solution.

Systems of Quadratic Equations

A system of equations is a set of equations with the same variables, and solving a system of quadratic equations involves finding the values of the variables that make all the equations true simultaneously.

Quadratic Inequalities

Quadratic inequalities involve comparing a quadratic function to another function or constant. Solving these inequalities involves isolating the variable on one side of the inequality and then determining the possible values of the variable.

In conclusion, quadratic equations and functions play a significant role in mathematics and are used to model various real-world scenarios. Understanding and solving quadratic equations involves various techniques and methods, including factorization, completing the square, and using the quadratic formula. Additionally, graphing quadratic functions and understanding their features are essential for working with these equations.