Mathematics
The Quadratic Formula and the Discriminant
The quadratic formula is a mathematical equation used to solve quadratic equations of the form ax^2 + bx + c = 0. It is given by x = (-b ± √(b^2 - 4ac)) / (2a). The discriminant, b^2 - 4ac, is a key component of the quadratic formula and determines the nature of the solutions (real, imaginary, or equal) based on its value.
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12 Key excerpts on "The Quadratic Formula and the Discriminant"
eBook - PDFAlgebra
A Combined Course 2E
- Charles P. McKeague(Author)
- 2018(Publication Date)
- XYZ Textbooks(Publisher)
For example, if we were to use the quadratic formula to solve the equation 2x 2 + 2x + 3 = 0, we would find the discriminant to be b 2 − 4ac = 2 2 − 4(2)(3) = −20 Because the discriminant appears under a square root symbol, we have the square root of a negative number in the quadratic formula. Our solutions would therefore be complex numbers. Similarly, if the discriminant were 0, the quadratic formula would yield x = −b ± √ — 0 ________ 2a = −b ± 0 ______ 2a = −b ___ 2a and the equation would have one rational solution, the number −b ___ 2a . ©iStockphoto.com/fredrocko Chapter 12 Quadratic Equations and Functions 942 The following table gives the relationship between the discriminant and the type of solutions to the equation. For the equation ax 2 + bx + c = 0 where a, b, and c are integers and a ≠ 0: In the second and third cases, when the discriminant is 0 or a positive perfect square, the solutions are rational numbers. The quadratic equations in these two cases are the ones that can be factored. EXAMPLE 1 For each equation, give the number and kind of solutions. a. x 2 − 3x − 40 = 0 Solution Using a = 1, b = −3, and c = −40 in b 2 − 4ac, we have (−3) 2 − 4(1)(−40) = 9 + 160 = 169. The discriminant is a perfect square. The equation therefore has two rational solutions. b. 2x 2 − 3x + 4 = 0 Solution Using a = 2, b = −3, and c = 4, we have b 2 − 4ac = (−3) 2 − 4(2)(4) = 9 − 32 = −23 The discriminant is negative, implying the equation has two complex solutions that contain i. c. 4x 2 − 12x + 9 = 0 Solution Using a = 4, b = −12, and c = 9, the discriminant is b 2 − 4ac = (−12) 2 − 4(4)(9) = 144 − 144 = 0 Because the discriminant is 0, the equation will have one rational solution. d. x 2 + 6x = 8 Solution We must first put the equation in standard form by adding −8 to each side. If we do so, the resulting equation is x 2 + 6x − 8 = 0 Now we identify a, b, and c as 1, 6, and −8, respectively.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Find the value of the discriminant. 2. Use the discriminant to determine the type of solutions a quadratic equation has. 3. Find a quadratic equation by reversing the factoring method. 4. Find a quadratic equation by reversing the Square Root Property for Equations method. More on Solutions to Quadratic Equations Introduction In this section, we will do two things. First, we will define the discriminant and use it to find the kind of solutions a quadratic equation has without solving the equation. Second, we will reverse the Zero-Factor Property or Square Root Property for Equations to build equations from their solutions. The Discriminant The quadratic formula x = − b ± √ — b 2 − 4 ac ______________ 2 a gives the solutions to any quadratic equation in standard form. There are times, when working with quadratic equations, that it is important only to know what kind of solutions the equation has. The discriminant indicates the number and type of solutions to a quadratic equation, when the original equation has integer coefficients. For example, if we were to use the quadratic formula to solve the equation 2 x 2 + 2 x + 3 = 0, we would find the discriminant to be b 2 − 4 ac = 2 2 − 4(2)(3) = − 20 Because the discriminant appears under a square root symbol, we have the square root of a negative number in the quadratic formula. Our solutions would therefore be complex numbers. Similarly, if the discriminant were 0, the quadratic formula would yield x = − b ± √ — 0 ________ 2 a = − b ± 0 ______ 2 a = − b ___ 2 a and the equation would have one rational solution, the number − b ___ 2 a . The following table gives the relationship between the discriminant and the type of solutions to the equation. For the equation ax 2 + bx + c = 0 where a , b , and c are integers and a ≠ 0: The expression under the radical in the quadratic formula is called the discriminant: Discriminant = D = b 2 − 4 ac discriminant DEFINITION
eBook - PDFIntermediate Algebra
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
©iStockphoto.com/fredrocko DEFINITION discriminant The expression under the radical in the quadratic formula is called the discriminant: discriminant = D = b 2 − 4ac OBJECTIVES KEY WORDS Chapter 7 Quadratic Equations and Functions 608 The following table gives the relationship between the discriminant and the type of solutions to the equation. For the equation ax 2 + bx + c = 0 where a, b, and c are integers and a ≠ 0: In the second and third cases, when the discriminant is 0 or a positive perfect square, the solutions are rational numbers. The quadratic equations in these two cases are the ones that can be factored. EXAMPLE 1 For each equation, give the number and kind of solutions. a. x 2 − 3x − 40 = 0 Solution Using a = 1, b = −3, and c = −40 in b 2 − 4ac, we have (−3) 2 − 4(1)(−40) = 9 + 160 = 169. The discriminant is a perfect square. The equation therefore has two rational solutions. b. 2x 2 − 3x + 4 = 0 Solution Using a = 2, b = −3, and c = 4, we have b 2 − 4ac = (−3) 2 − 4(2)(4) = 9 − 32 = −23 The discriminant is negative, implying the equation has two complex solutions that contain i. c. 4x 2 − 12x + 9 = 0 Solution Using a = 4, b = −12, and c = 9, the discriminant is b 2 − 4ac = (−12) 2 − 4(4)(9) = 144 − 144 = 0 Because the discriminant is 0, the equation will have one rational solution. d. x 2 + 6x = 8 Solution We must first put the equation in standard form by adding −8 to each side. If we do so, the resulting equation is x 2 + 6x − 8 = 0 Now we identify a, b, and c as 1, 6, and −8, respectively. b 2 − 4ac = 6 2 − 4(1)(−8) = 36 + 32 = 68 The discriminant is a positive number, but not a perfect square. The equation will therefore have two irrational solutions. B Finding an Unknown Constant EXAMPLE 2 Find an appropriate k so that the equation 4x 2 − kx = −9 has exactly one rational solution. Solution We begin by writing the equation in standard form.
eBook - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Verify the solutions of a quadratic equation by showing that the sum of the solutions is 2 b a and the product is c a . 2 3 1 Section The Discriminant and Equations That Can Be Written in Quadratic Form 8.3 Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2013 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. 554 CHAPTER 8 Quadratic Equations and Functions; Inequalities; and Algebra, Composition, and Inverses of Functions Vocabulary discriminant Getting Ready Evaluate b 2 2 4 ac for the following values. 1. a 5 2 , b 5 3 , and c 5 2 1 2. a 5 2 2 , b 5 4 , and c 5 2 3 Recall that we can test a trinomial of the form ax 2 1 bx 1 c 1 a 2 0 2 for factorability by evaluating b 2 2 4 ac . If b 2 2 4 ac is a perfect square, the trinomial is factorable. We rec-ognize that b 2 2 4 ac is the radicand in the quadratic formula. We can use that part of the quadratic formula to determine the type of solutions that a quadratic equation will have without solving it first. Use the discriminant to determine the type of solutions to a given quadratic equation. Suppose that the coefficients a , b , and c in the equation ax 2 1 bx 1 c 5 0 1 a 2 0 2 are real numbers. Then the solutions of the equation are given by the quadratic formula x 5 2 b 6 b 2 2 4 ac 2 a 1 a 2 0 2 The value of b 2 2 4 ac , called the discriminant , determines the type of solutions for any quadratic equation. Note that b 2 2 4 ac is a radicand; thus if b 2 2 4 ac $ 0 , the solutions are real numbers and if b 2 2 4 ac , 0 , the solutions are nonreal complex numbers.
eBook - PDFThe Algebra Teacher's Guide to Reteaching Essential Concepts and Skills
150 Mini-Lessons for Correcting Common Mistakes
- Judith A. Muschla, Gary Robert Muschla, Erin Muschla(Authors)
- 2011(Publication Date)
- Jossey-Bass(Publisher)
Students often make computational mistakes when finding the discriminant or they confuse the number of solutions if the discriminant is positive, negative, or zero. 1. Review the quadratic formula, x = − b ± √ b 2 − 4 ac 2 a , with your students. 2. Review the information and examples on the worksheet with your students. Emphasize that the expression inside the radical symbol, b 2 − 4 ac , is called the discriminant and can be evaluated by using the order of operations. Its value indicates the number of solutions to a quadratic equation. Note that the examples are written in standard form. Also explain the following: • There is one solution when the discriminant is 0. When 0 is substituted for the discrimi-nant in the quadratic formula, − b 2 a has only one value. • There are two solutions if the discriminant is positive: the quantity − b plus the square root of the discriminant divided by 2 a and the quantity − b minus the square root of the discriminate divided by 2 a . • There are no real solutions if the discriminant is negative. The square root of a negative number is not a real number. EXTRA HELP: Be sure that the equation is in standard form before you find the value of the discriminant. ANSWER KEY: (1) Two solutions (2) No real solutions (3) Two solutions (4) One solution ------------------------------------------------------------------------------------------(Challenge) It is important to find the discriminant before solving a quadratic equation because the discriminant allows you to know how many solutions you are looking for. ------------------------------------------------------------------------------------------220 T H E A L G E B R A T E A C H E R ’ S G U I D E Name Date WORKSHEET 5.23: USING THE DISCRIMINANT -------------------------------------------------------------------------------------The discriminant, b 2 − 4 ac , is the expression that is inside the radical symbol in the quadratic formula, x = − b ± √ b 2 − 4 ac 2 a .
eBook - PDF- R. Gustafson, Jeff Hughes(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Use the Quadratic Formula. Solve: 3x 2 2 x 2 5 5 0. x 5 2b 6 Ïb 2 2 4ac 2a x 5 2s21d 6 Ïs 21d 2 2 4s3ds 25d 2s3d a 53 b 52 1 c 52 5 x 5 1 6 Ï1 1 60 6 x 5 1 6 Ï61 6 Copyright 2017 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. Section 1.4 Quadratic Equations 129 t 5 88 6 Ï7744 2 64h 32 Simplify. 6. Define and Use the Discriminant We can predict the number and type of solutions or roots a quadratic equation will have before we solve it. Suppose that the coefficients a, b, and c in the equation ax 2 1 bx 1 c 5 0 sa Þ 0d are real numbers. Then the two roots of the equation are given by the Quadratic Formula x 5 2b 6 Ïb 2 2 4ac 2a sa Þ 0d The value of b 2 2 4ac, called the discriminant, determines the number and nature of the roots. The possibilities are summarized in the table as follows. Discriminant Number and Type of Roots 0 One repeated rational number Positive and a perfect square Two different rational numbers Positive and not a perfect square Two different irrational numbers Negative Two different nonreal complex numbers Take Note The discriminant is b 2 2 4ac and not Ïb 2 2 4ac . EXAMPLE 9 Using the Discriminant to Determine the Number and Type of Roots of a Quadratic Equation Determine the number and type of roots of 3x 2 1 4x 1 1 5 0. We calculate the discriminant b 2 2 4ac. b 2 2 4ac 5 4 2 2 4s3ds1d Substitute 4 for b, 3 for a, and 1 for c. 5 16 2 12 5 4 Since a, b, and c are real numbers and the discriminant is positive and a perfect square, then the roots will be two different rational numbers.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Our solutions would therefore be complex numbers. Similarly, if the discriminant were 0, the quadratic formula would yield x = − b ± √ — 0 ________ 2 a = − b ± 0 ______ 2 a = − b ___ 2 a and the equation would have one rational solution, the number − b ___ 2 a . The following table gives the relationship between the discriminant and the type of solutions to the equation. For the equation ax 2 + bx + c = 0 where a , b , and c are integers and a ≠ 0: The expression under the radical in the quadratic formula is called the discriminant: Discriminant = D = b 2 − 4 ac discriminant DEFINITION 800 Chapter 11 Quadratic Equations and Functions In the first case, when a quadratic equation has non-real complex solutions, they will always be complex conjugates. In the second and third cases, when the discriminant is 0 or a positive perfect square, the solutions are rational numbers. The quadratic equations in these two cases are the ones that can be factored. For each equation, give the number and kind of solutions. 1. x 2 − 3 x − 40 = 0 SOLUTION Using a = 1, b = − 3, and c = − 40 in b 2 − 4 ac , we have ( − 3) 2 − 4(1)( − 40) = 9 + 160 = 169. The discriminant is a perfect square. The equation therefore has two rational solutions. 2. 2 x 2 − 3 x + 4 = 0 SOLUTION Using a = 2, b = − 3, and c = 4, we have b 2 − 4 ac = ( − 3) 2 − 4(2)(4) = 9 − 32 = − 23 The discriminant is negative, implying the equation has two complex solutions that contain i . 3. 4 x 2 − 12 x + 9 = 0 SOLUTION Using a = 4, b = − 12, and c = 9, the discriminant is b 2 − 4 ac = ( − 12) 2 − 4(4)(9) = 144 − 144 = 0 Because the discriminant is 0, the equation will have one rational solution. 4. x 2 + 6 x = 8 SOLUTION We must first put the equation in standard form by adding − 8 to each side. If we do so, the resulting equation is x 2 + 6 x − 8 = 0 Now we identify a , b , and c as 1, 6, and − 8, respectively: b 2 − 4 ac = 6 2 − 4(1)( − 8) = 36 + 32 = 68 The discriminant is a positive number, but not a perfect square.
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
TRY IT : : 10.70 Solve 25t 2 − 40t = −16 by using the Quadratic Formula. Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions. Is there a way to predict the number of solutions to a quadratic equation without actually solving the equation? Yes, the quantity inside the radical of the Quadratic Formula makes it easy for us to determine the number of solutions. This quantity is called the discriminant. Discriminant In the Quadratic Formula x = −b ± b 2 − 4ac 2a , the quantity b 2 − 4ac is called the discriminant. Let’s look at the discriminant of the equations in Example 10.28, Example 10.32, and Example 10.35, and the number of solutions to those quadratic equations. Quadratic Equation (in standard form) Discriminant b 2 − 4ac Sign of the Discriminant Number of real solutions Example 10.28 2x 2 + 9x − 5 = 0 9 2 − 4 · 2(−5) = 121 + 2 Example 10.35 4x 2 − 20x + 25 = 0 (−20) 2 − 4 · 4 · 25 = 0 0 1 Example 10.32 3 p 2 + 2 p + 9 = 0 2 2 − 4 · 3 · 9 = −104 − 0 Chapter 10 Quadratic Equations 1191 When the discriminant is positive ⎛ ⎝ x = −b ± + 2a ⎞ ⎠ the quadratic equation has two solutions. When the discriminant is zero ⎛ ⎝ x = −b ± 0 2a ⎞ ⎠ the quadratic equation has one solution. When the discriminant is negative ⎛ ⎝ x = −b ± − 2a ⎞ ⎠ the quadratic equation has no real solutions. EXAMPLE 10.36 Determine the number of solutions to each quadratic equation: ⓐ 2v 2 − 3v + 6 = 0 ⓑ 3x 2 + 7x − 9 = 0 ⓒ 5n 2 + n + 4 = 0 ⓓ 9y 2 − 6y + 1 = 0 Solution To determine the number of solutions of each quadratic equation, we will look at its discriminant. ⓐ 2v 2 − 3v + 6 = 0 The equation is in standard form, identify a, b, c. a = 2, b = −3, c = 6 Write the discriminant. b 2 − 4ac Substitute in the values of a, b, c.
eBook - PDFThe Higher Arithmetic
An Introduction to the Theory of Numbers
- H. Davenport(Author)
- 2008(Publication Date)
- Cambridge University Press(Publisher)
It is a purely algebraical rela- tion, and we have here a particular instance of a very general situation. A function of the coefficients of an algebraic form, such as b 2 − 4ac in the present case, which is unaltered when a linear substitution of determinant 1 is applied to the form, is said to be an algebraic invariant of the form. The discriminant of a binary quadratic form is a simple example of such an invariant. Although equivalent forms have the same discriminant, it is by no means true that forms of the same discriminant are necessarily equivalent. For example, the forms (1, 0, 6) and (2, 0, 3) both have the discriminant −24, but they are not equivalent. To see this, we need only observe that the form x 2 + 6 y 2 represents the number 1, namely when x = 1 and y = 0, whereas the form 2x 2 + 3 y 2 can obviously never take the value 1. The discriminant d of a quadratic form is an integer, which may be pos- itive, negative or zero. Not every integer can figure as the discriminant of a form. For b 2 − 4ac ≡ b 2 (mod 4), and any square is congruent to 0 or 1 Quadratic Forms 121 (mod 4). Hence d must be congruent to 0 or 1 (mod 4), and the possible discriminants are . . . , −11, −8, −7, −4, −3, 0, 1, 4, 5, 8, 9, . . . . Moreover, each such number is the discriminant of at least one form. For if d is any given number which is congruent to 0 or 1 (mod 4), we can satisfy the equation b 2 − 4ac = d by taking a to be 1, and taking b to be 0 or 1 according as d ≡ 0 or 1 (mod 4). Then c is − 1 4 d or − 1 4 (d − 1), as the case may be. This gives a particular form of discriminant d , namely 1, 0, − 1 4 d or 1, 1, − 1 4 (d − 1) according as d ≡ 0 or 1 (mod 4). This is called the principal form of discriminant d . Thus the principal form of discriminant −4 is (1, 0, 1), or x 2 + y 2 , and the principal form of discriminant 5 is (1, 1, −1), or x 2 + xy − y 2 .
eBook - PDFQuadratic Irrationals
An Introduction to Classical Number Theory
- Franz Halter-Koch(Author)
- 2013(Publication Date)
- Chapman and Hall/CRC(Publisher)
An integer Δ ∈ Z is called a quadratic discriminant if Δ is not a square and Δ ≡ 0 or 1 mod 4. If ξ is a quadratic irrational of type ( a, b, c ) and discriminant Δ = b 2 − 4 ac , then Δ is a quadratic discriminant according to this definition. For a quadratic discriminant Δ, we define ω Δ = σ Δ + √ Δ 2 , where σ Δ = 0 if Δ ≡ 0 mod 4 , 1 if Δ ≡ 1 mod 4 . If Δ = 4 D + σ Δ , where D ∈ Z , then ω Δ is a quadratic irrational of type (1 , σ Δ , − D ) and discriminant Δ, ω Δ = − ω Δ + σ Δ , and ω 2 Δ = D + σ Δ ω Δ . The quadratic irrational ω Δ is called the basis number of discriminant Δ. If Δ is a quadratic discriminant and f ∈ N , then Δ f 2 is also a quadratic discrimi-nant (since f 2 ≡ 0 or 1 mod 4). A quadratic discriminant Δ is called a fundamental discriminant if it is not of the form Δ = Δ 1 f 2 for some quadratic discriminant Δ 1 and an integer f ≥ 2. Theorem 1.1.6. 1. For a quadratic discriminant Δ , the following assertions are equivalent : (a) Δ is a fundamental discriminant. (b) Either Δ is squarefree and Δ ≡ 1 mod 4 , or Δ = 4 D for some square-free integer D such that D ≡ 2 or 3 mod 4 . (c) v p (Δ) ∈ { 0 , 1 } for all odd primes p , and Δ ≡ 1 mod 4 or Δ ≡ 8 mod 16 or Δ ≡ 12 mod 16 . 1.1. QUADRATIC IRRATIONALS, QUADRATIC NUMBER FIELDS AND DISCRIMINANTS 5 2. Let d ∈ Q be not a square. Then there is a unique fundamental discriminant Δ 0 such that d = Δ 0 q 2 for some q ∈ Q > 0 , and if d itself is a quadratic discriminant, then q ∈ N . 3. Every quadratic irrational has a unique representation ξ = u + v √ Δ 0 , where u ∈ Q , v ∈ Q × , and Δ 0 is a fundamental discriminant. In this case, there is some f ∈ N such that Δ 0 f 2 is the discriminant of ξ . Proof. We apply Theorem A.3.3. 1. (a) ⇒ (b) CASE 1 : Δ ≡ 1 mod 4. If Δ is not squarefree, then Δ = Δ 1 f 2 for some Δ 1 ∈ Z and some odd integer f ≥ 2. Then Δ 1 ≡ 1 mod 4 is not a square and thus a quadratic discriminant, a contradiction. CASE 2 : Δ = 4 D for some D ∈ Z which is not a square.
eBook - PDFAlgebra
Form and Function
- William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Solving Quadratic Equations with Perfect Squares Factoring is a useful technique for solving quadratic equations, but sometimes factoring may be difficult (or impossible). For example, in order to use factoring to solve the quadratic equation ( − 2) 2 = 5, we need to first write the equation in the standard form by expanding the left-hand side of the equation and subtracting 5 from both sides. This gives the equation 2 − 4 − 1 = 0, which cannot be factored with integer coefficients. Instead, since we have a perfect square on the left-hand side of the equation and a positive value on the right-hand side, we can take the square root of both sides to find solutions more quickly: ( − 2) 2 = 5 − 2 = ± √ 5 take square root of both sides 6 = 2 ± √ 5. add 2 to both sides 6 See page 375 in Appendix D for a review of why taking the square root of both sides results in two solutions. 118 Chapter 3 QUADRATIC FUNCTIONS In general, if we can put a quadratic equation in the form ( − ℎ) 2 = , where ℎ and are constants, with ≥ 0, then we can solve it by taking square roots of both sides. Example 2 Solve the quadratic equation by taking square roots of both sides (if possible). (a) 2 − 4 = 0 (b) 2 − 5 = 0 (c) 2( + 1) 2 = 0 (d) 2( − 3) 2 + 4 = 0. Solution (a) Rewriting the equation as 2 = 4, we see the solutions are = ± √ 4 = ±2. (b) Similarly, the solutions to 2 − 5 = 0 are = ± √ 5 = ±2.236. (c) Since 2( + 1) 2 = 0, dividing by 2 gives ( + 1) 2 = 0 so + 1 = 0. Thus the only solution is = −1. It is possible for a quadratic equation to have only one solution. (d) Since 2( − 3) 2 + 4 = 0, we have 2( − 3) 2 = −4, so dividing by 2 gives ( − 3) 2 = −2. But since no real number squared is −2, this equation has no real solutions. Example 2 illustrates that it is possible for a quadratic equation to have two solutions, just one solution, or no solution.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2017(Publication Date)
- XYZ Textbooks(Publisher)
My life was actually going in that direction so I decided to see where that would take me. It was a good decision. It is a good idea to work hard toward your goals, but it is also a good idea to take inventory every now and then to be sure you are headed in the direction that is best for you. I wish you good luck with the rest of your college years, and with whatever you decide you want to do as a career. Pat McKeague 619 9.5 Linear Equations Learning Objectives Learning Objectives In this section, we will learn how to: 1. Write square roots of negative numbers in terms of i . 2. Solve quadratic equations having complex solutions. 9.5 Introduction The quadratic formula tells us that the solutions to equations of the form ax 2 + bx + c = 0 are always: x = − b ± √ — b 2 − 4 ac ______________ 2 a The part of the quadratic formula under the radical sign is called the discriminant : Discriminant = b 2 − 4 ac When the discriminant is negative, we have to deal with the square root of a negative number. We handle square roots of negative numbers by using the definition i = √ — − 1. To illustrate, suppose we want to simplify an expression that contains √ — − 9, which is not a real number. We begin by writing √ — − 9 as √ — 9( − 1) . Then, we write this expression as the product of two separate radicals: √ — 9 ∙ √ — − 1 . Applying the definition i = √ — − 1 to this last expression, we have √ — 9 ∙ √ — − 1 = 3 i As you may recall from the previous section, the number 3 i is called a complex number. Here are some further examples. Write the following radicals as complex numbers: SOLUTION a. √ — − 4 = √ — 4( − 1) = √ — 4 ∙ √ — − 1 = 2 i b. √ — − 36 = √ — 36( − 1) = √ — 36 ∙ √ — − 1 = 6 i c. √ — − 7 = √ — 7( − 1) = √ — 7 ∙ √ — − 1 = i √ — 7 d. √ — − 75 = √ — 75( − 1) = √ — 75 ∙ √ — − 1 = 5 i √ — 3 In parts (c) and (d) of Example 1, we wrote i before the radical because it is less confusing that way.
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