Skip to Content
OpenStax Logo
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

9.1 Solve Quadratic Equations Using the Square Root Property

  • Square Root Property
    • If x2=kx2=k, then x=korx=kx=korx=k or x=±kx=±k

    How to solve a quadratic equation using the square root property.
    1. Step 1. Isolate the quadratic term and make its coefficient one.
    2. Step 2. Use Square Root Property.
    3. Step 3. Simplify the radical.
    4. Step 4. Check the solutions.

9.2 Solve Quadratic Equations by Completing the Square

  • Binomial Squares Pattern
    If a and b are real numbers,
    Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.
  • How to Complete a Square
    1. Step 1. Identify b, the coefficient of x.
    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 + bx
    4. Step 4. Rewrite the trinomial as a binomial square
  • How to solve a quadratic equation of the form ax2 + bx + c = 0 by completing the square.
    1. Step 1. Divide by a to make the coefficient of x2 term 1.
    2. Step 2. Isolate the variable terms on one side and the constant terms on the other.
    3. Step 3. Find (12·b)2,(12·b)2, the number needed to complete the square. Add it to both sides of the equation.
    4. Step 4. Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right.
    5. Step 5. Use the Square Root Property.
    6. Step 6. Simplify the radical and then solve the two resulting equations.
    7. Step 7. Check the solutions.

9.3 Solve Quadratic Equations Using the Quadratic Formula

  • Quadratic Formula
    • The solutions to a quadratic equation of the form ax2 + bx + c = 0, a0a0 are given by the formula:
      x=b±b24ac2ax=b±b24ac2a
  • How to solve a quadratic equation using the Quadratic Formula.
    1. Step 1. Write the quadratic equation in standard form, ax2 + bx + c = 0. Identify the values of a, b, c.
    2. Step 2. Write the Quadratic Formula. Then substitute in the values of a, b, c.
    3. Step 3. Simplify.
    4. Step 4. Check the solutions.
  • Using the Discriminant, b2 − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation
    • For a quadratic equation of the form ax2 + bx + c = 0, a0,a0,
      • If b2 − 4ac > 0, the equation has 2 real solutions.
      • if b2 − 4ac = 0, the equation has 1 real solution.
      • if b2 − 4ac < 0, the equation has 2 complex solutions.
  • Methods to Solve Quadratic Equations:
    • Factoring
    • Square Root Property
    • Completing the Square
    • Quadratic Formula
  • How 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 = k or a(xh)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.

9.4 Solve Quadratic Equations in Quadratic Form

  • How to solve equations in quadratic form.
    1. Step 1. Identify a substitution that will put the equation in quadratic form.
    2. Step 2. Rewrite the equation with the substitution to put it in quadratic form.
    3. Step 3. Solve the quadratic equation for u.
    4. Step 4. Substitute the original variable back into the results, using the substitution.
    5. Step 5. Solve for the original variable.
    6. Step 6. Check the solutions.

9.5 Solve Applications of Quadratic Equations

  • Methods to Solve Quadratic Equations
    • Factoring
    • Square Root Property
    • Completing the Square
    • Quadratic Formula
  • How to use a Problem-Solving Strategy.
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
  • Area of a Triangle
    • For a triangle with base, b, and height, h, the area, A, is given by the formula A=12bh.A=12bh.
      Image of a trangle. The horizontal base side is labeled b, and a line segment labeled h is perpendicular to the base, connecting it to the opposite vertex.
  • Area of a Rectangle
    • For a rectangle with length, L, and width, W, the area, A, is given by the formula A = LW.
      Image shows a rectangle. All four angles are marked as right angles. The longer, horizontal side is labeled L and the shorter, vertical side is labeled w.
  • Pythagorean Theorem
    • In any right triangle, where a and b are the lengths of the legs, and c is the length of the hypotenuse, a2 + b2 = c2.
      Image shows a right triangle with horizontal and vertical legs. The vertical leg is labeled a. The horizontal side is labeled b. The hypotenuse is labeled c.
  • Projectile motion
    • The height in feet, h, of an object shot upwards into the air with initial velocity, v0, after t seconds is given by the formula h = −16t2 + v0t.

9.6 Graph Quadratic Functions Using Properties

  • Parabola Orientation
    • For the graph of the quadratic function f(x)=ax2+bx+c,f(x)=ax2+bx+c, if
      • a > 0, the parabola opens upward.
      • a < 0, the parabola opens downward.
  • Axis of Symmetry and Vertex of a Parabola The graph of the function f(x)=ax2+bx+cf(x)=ax2+bx+c is a parabola where:
    • the axis of symmetry is the vertical line x=b2a.x=b2a.
    • the vertex is a point on the axis of symmetry, so its x-coordinate is b2a.b2a.
    • the y-coordinate of the vertex is found by substituting x=b2ax=b2a into the quadratic equation.
  • Find the Intercepts of a Parabola
    • To find the intercepts of a parabola whose function is f(x)=ax2+bx+c:f(x)=ax2+bx+c:
      y-interceptx-interceptsLetx=0and solve forf(x).Letf(x)=0and solve forx.y-interceptx-interceptsLetx=0and solve forf(x).Letf(x)=0and solve forx.
  • How to graph a quadratic function using properties.
    1. Step 1. Determine whether the parabola opens upward or downward.
    2. Step 2. Find the equation of the axis of symmetry.
    3. Step 3. Find the vertex.
    4. Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
    5. Step 5. Find the x-intercepts. Find additional points if needed.
    6. Step 6. 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.

9.7 Graph Quadratic Functions Using Transformations

  • Graph a Quadratic Function of the form f(x)=x2+kf(x)=x2+k Using a Vertical Shift
    • The graph of f(x)=x2+kf(x)=x2+k shifts the graph of f(x)=x2f(x)=x2 vertically k units.
      • If k > 0, shift the parabola vertically up k units.
      • If k < 0, shift the parabola vertically down |k||k| units.
  • Graph a Quadratic Function of the form f(x)=(xh)2f(x)=(xh)2 Using a Horizontal Shift
    • The graph of f(x)=(xh)2f(x)=(xh)2 shifts the graph of f(x)=x2f(x)=x2 horizontally h units.
      • If h > 0, shift the parabola horizontally left h units.
      • If h < 0, shift the parabola horizontally right |h||h| units.
  • Graph of a Quadratic Function of the form f(x)=ax2f(x)=ax2
    • The coefficient a in the function f(x)=ax2f(x)=ax2 affects the graph of f(x)=x2f(x)=x2 by stretching or compressing it.
      If 0<|a|<1,0<|a|<1, then the graph of f(x)=ax2f(x)=ax2 will be “wider” than the graph of f(x)=x2.f(x)=x2.
      If |a|>1,|a|>1, then the graph of f(x)=ax2f(x)=ax2 will be “skinnier” than the graph of f(x)=x2.f(x)=x2.
  • How to graph a quadratic function using transformations
    1. Step 1. Rewrite the function in f(x)=a(xh)2+kf(x)=a(xh)2+k form by completing the square.
    2. Step 2. Graph the function using transformations.
  • Graph a quadratic function in the vertex form f(x)=a(xh)2+kf(x)=a(xh)2+k using properties
    1. Step 1. Rewrite the function in f(x)=a(xh)2+kf(x)=a(xh)2+k form.
    2. Step 2. Determine whether the parabola opens upward, a > 0, or downward, a < 0.
    3. Step 3. Find the axis of symmetry, x = h.
    4. Step 4. Find the vertex, (h, k).
    5. Step 5. Find they-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
    6. Step 6. Find the x-intercepts, if possible.
    7. Step 7. Graph the parabola.

9.8 Solve Quadratic Inequalities

  • Solve a Quadratic Inequality Graphically
    1. Step 1. Write the quadratic inequality in standard form.
    2. Step 2. Graph the function f(x)=ax2+bx+cf(x)=ax2+bx+c using properties or transformations.
    3. Step 3. Determine the solution from the graph.
  • How to Solve a Quadratic Inequality Algebraically
    1. Step 1. Write the quadratic inequality in standard form.
    2. Step 2. Determine the critical points -- the solutions to the related quadratic equation.
    3. Step 3. Use the critical points to divide the number line into intervals.
    4. Step 4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.
    5. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
Citation information

© Apr 15, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.