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Elementary Algebra

Key Concepts

Elementary AlgebraKey Concepts
Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Key Concepts

10.1 Solve Quadratic Equations Using the Square Root Property

  • Square Root Property
    If x2=kx2=k, and k≥0k≥0, then x=korx=−kx=korx=−k.

10.2 Solve Quadratic Equations by Completing the Square

  • Binomial Squares Pattern If a,ba,b are real numbers,
     (a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2
    (a−b)2=a2−2ab+b2(a−b)2=a2−2ab+b2
  • Complete a Square
    To complete the square of x2+bxx2+bx:
    1. Step 1. Identify bb, the coefficient of xx.
    2. Step 2. Find (12b)2(12b)2, the number to complete the square.
    3. Step 3. Add the (12b)2(12b)2 to x2+bxx2+bx.

10.3 Solve Quadratic Equations Using the Quadratic Formula

  • Quadratic Formula The solutions to a quadratic equation of the form ax2+bx+c=0,ax2+bx+c=0, a≠0a≠0 are given by the formula:
    x=−b±b2−4ac2ax=−b±b2−4ac2a
  • Solve a Quadratic Equation Using the Quadratic Formula
     To solve a quadratic equation using the Quadratic Formula.
    1. Step 1. Write the quadratic formula in standard form. Identify the a,b,ca,b,c values.
    2. Step 2. Write the quadratic formula. Then substitute in the values of a,b,c.a,b,c.
    3. Step 3. Simplify.
    4. Step 4. Check the solutions.
  • Using the Discriminant, b2−4acb2−4ac, to Determine the Number of Solutions of a Quadratic Equation
     For a quadratic equation of the form ax2+bx+c=0,ax2+bx+c=0, a≠0,a≠0,
    • if b2−4ac>0b2−4ac>0, the equation has 2 solutions.
    • if b2−4ac=0b2−4ac=0, the equation has 1 solution.
    • if b2−4ac<0b2−4ac<0, the equation has no real solutions.
  • To identify the most appropriate method to solve a quadratic equation:
    1. Step 1. Try Factoring first. If the quadratic factors easily this method is very quick.
    2. Step 2. Try the Square Root Property next. If the equation fits the form ax2=kax2=k or a(x−h)2=ka(x−h)2=k, it can easily be solved by using the Square Root Property.
    3. Step 3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

10.4 Solve Applications Modeled by Quadratic Equations

  • Area of a Triangle For a triangle with base, bb, and height, hh, the area, AA, is given by the formula: A=12bhA=12bh
  • Pythagorean Theorem In any right triangle, where aa and bb are the lengths of the legs, and cc is the length of the hypothenuse, a2+b2=c2a2+b2=c2
  • Projectile motion The height in feet, hh, of an object shot upwards into the air with initial velocity, v0v0, after tt seconds can be modeled by the formula:
    h=−16t2+v0th=−16t2+v0t

10.5 Graphing Quadratic Equations

  • The graph of every quadratic equation is a parabola.
  • Parabola Orientation For the quadratic equation y=ax2+bx+cy=ax2+bx+c, if
    • a>0a>0, the parabola opens upward.
    • a<0a<0, the parabola opens downward.
  • Axis of Symmetry and Vertex of a Parabola For a parabola with equation y=ax2+bx+cy=ax2+bx+c:
    • The axis of symmetry of a parabola is the line x=−b2ax=−b2a.
    • The vertex is on the axis of symmetry, so its x-coordinate is −b2a−b2a.
    • To find the y-coordinate of the vertex we substitute x=−b2ax=−b2a into the quadratic equation.
  • Find the Intercepts of a Parabola To find the intercepts of a parabola with equation y=ax2+bx+cy=ax2+bx+c:
     y-interceptx-interceptsLetx=0and solve fory.Lety=0and solve forx.y-interceptx-interceptsLetx=0and solve fory.Lety=0and solve forx.
  • To Graph a Quadratic Equation in Two Variables
    1. Step 1. Write the quadratic equation with yy on one side.
    2. Step 2. Determine whether the parabola opens upward or downward.
    3. Step 3. Find the axis of symmetry.
    4. Step 4. Find the vertex.
    5. Step 5. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
    6. Step 6. Find the x-intercepts.
    7. Step 7. Graph the parabola.
  • Minimum or Maximum Values of a Quadratic Equation
    • The y-coordinate of the vertex of the graph of a quadratic equation is the
    • minimum value of the quadratic equation if the parabola opens upward.
    • maximum value of the quadratic equation if the parabola opens downward.
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