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Elementary Algebra
Unit 7 Overview: Factoring
7.1 Greatest Common Factor and Factor by Grouping
7.2 Factor Trinomials of the Form x2+bx+c
7.3 Factor Trinomials of the Form ax2+bx+c
7.4 Factor Special Products
7.5 General Strategy for Factoring Polynomials
7.6 Quadratic Equations

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7.6 Quadratic Equations

Last Updated on June 25, 2024

Quadratic equations are powerful tools for solving real-world problems. They model situations like projectile motion and profit optimization. Understanding how to factor and solve these equations is key to unlocking their potential.

The Zero Product Property and factoring techniques are essential for solving quadratic equations. Graphing parabolas helps visualize solutions, while the quadratic formula provides a reliable method when factoring fails. These skills open doors to advanced problem-solving.

Quadratic Equations

Zero Product Property application

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  • States if the product of factors is zero, then at least one factor must be zero
    • Example: If ab=0ab = 0, then either a=0a = 0, b=0b = 0, or both are zero
  • Solving quadratic equations using Zero Product Property:
    1. Set quadratic expression equal to zero
    2. Factor the quadratic expression into its component factors
    3. Set each factor equal to zero and solve for the variable
    4. Solutions are values that make each factor equal zero (roots)

Factoring quadratic expressions

  • Rewriting polynomial as product of factors
  • Factoring quadratic expression ax2+bx+cax^2 + bx + c:
    • Find two numbers with product acac and sum bb
    • Rewrite quadratic expression using these numbers
    • Factor by grouping if necessary (splitting middle term)
  • After factoring, set each factor to zero and solve for variable to find solutions
    • Quadratic formula can factor when other methods fail: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
    • The expression under the square root (b2−4acb^2 - 4ac) is called the discriminant

Real-world quadratic modeling

  • Quadratic equations model various situations:
    • Height of object thrown upward (projectile motion)
    • Area of rectangular space (optimization)
    • Profit of business (revenue and cost functions)
  • Solving real-world problems with quadratic equations:
    1. Identify given information and unknown variable
    2. Create quadratic equation representing the situation
    3. Solve quadratic equation by factoring or Zero Product Property
    4. Interpret solutions in context of real-world problem
      • Determine which solutions are relevant and meaningful for situation
      • Example: Negative time values not applicable in projectile motion

Graphical representation of quadratic equations

  • The graph of a quadratic equation forms a parabola
  • Key features of a parabola:
    • Vertex: The highest or lowest point of the parabola
    • Axis of symmetry: A vertical line passing through the vertex
    • Roots: The x-intercepts of the parabola (solutions to the quadratic equation)
  • Completing the square: A method to rewrite a quadratic equation in vertex form, useful for finding the vertex and axis of symmetry

Key Terms to Review (17)

Factoring: Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental concept in algebra that is essential for solving a wide range of problems, including solving equations, simplifying rational expressions, and finding the roots of quadratic functions.
Polynomial: A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers. Polynomials are fundamental building blocks in algebra and are central to many topics in elementary algebra.
Factor by Grouping: Factor by grouping is a technique used to factor polynomials by identifying common factors among groups of terms and then factoring out those common factors. This method is particularly useful when dealing with polynomials that have more than two terms and where the traditional method of factoring by finding the greatest common factor (GCF) may not be sufficient.
Zero Product Property: The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This principle is fundamental in the process of factoring polynomials and solving equations involving products.
Roots: Roots, in the context of mathematics, refer to the solutions or values of a variable that satisfy an equation. They are the points where a function or equation intersects the x-axis, indicating the values of the independent variable that make the function or equation equal to zero.
Quadratic Equations: A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one variable with an exponent of 2. These equations are characterized by the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in algebra and have numerous applications in various fields, including physics, engineering, and economics.
Factors: Factors are the individual elements or components that contribute to or influence a particular outcome or phenomenon. In the context of quadratic equations, factors refer to the variables, coefficients, and constants that make up the equation and determine its behavior.
Discriminant: The discriminant is a mathematical expression that provides important information about the nature and characteristics of quadratic equations. It is a crucial concept in the study of quadratic functions and their solutions.
Quadratic Expressions: A quadratic expression is a polynomial expression that contains a variable raised to the second power, along with other terms involving the variable and constant terms. These expressions are fundamental in the study of quadratic equations, which are essential in various areas of mathematics and science.
$ax^2 + bx + c$: $ax^2 + bx + c$ is a general quadratic equation, where $a$, $b$, and $c$ are constants. This type of equation is fundamental in the study of quadratic functions and their applications in various mathematical and scientific fields.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. The formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Projectile Motion: Projectile motion is the motion of an object that is launched or thrown into the air and moves solely under the influence of gravity, without any additional propulsive force acting on it. This concept is central to understanding various applications in physics, including the trajectories of objects such as balls, rockets, and even the motion of celestial bodies.
Parabola: A parabola is a U-shaped curve that is the graph of a quadratic function. It is a fundamental shape in mathematics, with applications in various fields such as physics, engineering, and even art. The parabola is closely related to the concept of quadratic equations, which are central to the topics covered in this chapter.
Completing the Square: Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square form. This technique allows for the application of the square root property or the quadratic formula to find the solutions to the equation.
Axis of Symmetry: The axis of symmetry is a line that divides a graph or equation into two equal halves, where each side is a reflection of the other. This concept is particularly important in the study of quadratic equations and their graphical representations.
Optimization: Optimization is the process of making something as effective or functional as possible, often by maximizing or minimizing certain parameters. In the context of quadratic equations, it involves finding the maximum or minimum values of a quadratic function, which can be visualized as identifying the vertex of a parabola. This concept is crucial in various applications such as economics, engineering, and physics, where determining optimal solutions is essential for success.
Vertex: The vertex is a significant point in a quadratic equation represented in the form of a parabola, where it serves as either the maximum or minimum point of the graph. This point is crucial for understanding the overall shape and direction of the parabola, as it determines where the graph changes direction, impacting the solutions to the equation and the graphing of its curve.
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