Mathematics
Completing the Square
Completing the square is a technique used in algebra to manipulate quadratic equations into a standard form. This form can be used to easily solve for the roots of the equation, or to graph the equation. The process involves adding and subtracting terms to create a perfect square trinomial.
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2 Key excerpts on "Completing the Square"
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
In this section, we will discuss a procedure that enables us to solve quadratic equations such as , whose left side is not a perfect-square trinomial. To make the left side a perfect-square trinomial, we will use a procedure called Completing the Square. x 2 4 x 3 x 2 16 x 64 2 ( x 1) 2 50 ( x 3) 2 36 1. Factor: 2. What is the coefficient of the middle term of ? 3. Find one-half of 12. Then square that result. 4. Find . Then square that result. 5. Solve for x . 6. Add: 2 37 4 x 6 2 7 1 2 3 x 2 6 x 14 x 2 10 x 25 The following problems review some basic skills that are needed when solving quadratic equations by Completing the Square. Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Complete the Square to Write Perfect-Square Trinomials. In Section 5.7, we learned how to square binomials quickly using special-product rules. Two examples are shown below. The square of a binomial The perfect-square trinomial result = = In both results, there is a relationship between the coefficient of x and the constant (third) term. In , for example, the coefficient of x is 8 and the constant term is 16. Note that the constant, 16, is the square of one-half the coefficient of x . Similarly, in , the constant, 25, is the square of one-half the coefficient of x , which is . Now, let's generalize. Consider the following perfect-square trinomials (with leading coefficients of 1) and their factored forms. In each of these perfect-square trinomials, the third term is the square of one-half of the coefficient of .
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
Chapter 10 Quadratic Equations 1231 Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? 1232 Chapter 10 Quadratic Equations This OpenStax book is available for free at http://cnx.org/content/col31130/1.4 axis of symmetry Completing the Square consecutive even integers consecutive odd integers discriminant parabola quadratic equation quadratic equation in two variables Square Root Property vertex x-intercepts of a parabola y-intercept of a parabola CHAPTER 10 REVIEW KEY TERMS The axis of symmetry is the vertical line passing through the middle of the parabola y = ax 2 + bx + c. Completing the Square is a method used to solve quadratic equations. Consecutive even integers are even integers that follow right after one another. If an even integer is represented by n , the next consecutive even integer is n + 2 , and the next after that is n + 4 . Consecutive odd integers are odd integers that follow right after one another. If an odd integer is represented by n , the next consecutive odd integer is n + 2 , and the next after that is n + 4 . In the Quadratic Formula, x = −b ± b 2 − 4ac 2a the quantity b 2 − 4ac is called the discriminant. The graph of a quadratic equation in two variables is a parabola. A quadratic equation is an equation of the form ax 2 + bx + c = 0 , where a ≠ 0. A quadratic equation in two variables, where a, b, and c are real numbers and a ≠ 0 is an equation of the form y = ax 2 + bx + c. The Square Root Property states that, if x 2 = k and k ≥ 0 , then x = k or x = − k. The point on the parabola that is on the axis of symmetry is called the vertex of the parabola; it is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. The x-intercepts are the points on the parabola where y = 0.
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