Quadratic Equations Study Guide

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form:

ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Ways to Solve Quadratic Equations

1. Factoring: If the quadratic equation can be factored, then we can set each factor equal to zero and solve for the variable.
2. Quadratic Formula: The quadratic formula is used to solve any quadratic equation and is given by:

x = (-b ± √(b² - 4ac)) / (2a)

3. Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be easily solved.

Graphical Representation of Quadratic Equations

The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the value of the coefficient a. The x-coordinate of the vertex of the parabola is given by -b/2a, and the y-coordinate can be found by substituting this x-value into the original equation.

Quadratic Equations in Real Life

Quadratic equations are widely used in physics, engineering, economics, and many other fields to model various real-life situations such as projectile motion, optimization problems, and financial analysis.

Practice Problems

Solve the following quadratic equations:

1. 2x² - 5x + 2 = 0
2. x² + 4x - 5 = 0