2.09 Quadratic Equations by Factoring from Basic Algebra: One Step at a Time © 2002 P. 185-192 Dr. Robert J. Rapalje Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
A great deal of effort has just been spent learning to factor algebraic expressions. The question has probably been raised more than once, "What good is factoring?" Many situations arise in math in which it is necessary to use the factored form of an expression. One such situation is solving a quadratic equation−−an equation in which the variable is raised to the second power. In general, the equation is in the form ax2+bx+c=0. Before solving quadratic equations by the method of factoring, it is necessary to remember one very important fact about numbers: If the product of two numbers is zero, then one of the numbers must be zero. First notice that this is a special property of zero, and there is no other number of which this could be said. Secondly, notice that in order to use this property, it is necessary to use a product (i.e., factored form) of numbers. Of course, this is where the process of factoring comes in.
EXAMPLE 1. Solve x2 + 3x = 0 Solution: Factor x(x + 3) = 0 This is a product equal to zero. x=0 or x+3 = 0 Set each factor equal to zero. −3 −3 Solve for x in each case. x = 0; x = −3 You may wish to check these answers in the original equation. Check: x = 0 x = − 3 02 +3(0) = 0 (−3)2 + 3(−3) = 0 0 = 0 9 + (−9) = 0
EXAMPLE 2. Solve x2 + 3x = 4 Solution: Notice that this equation does not equal zero. The first step is to make it equal to zero by adding −4 to each side: x2 + 3x − 4 = 0 Factor: (x + 4)(x − 1) = 0 At this point it may be helpful to think of splitting the problem in half. Think of exchanging the one "large" equation for two "small" ones, as shown below:
−4 −4 +1 +1 Solve for x in each case. x = −4 x = 1 Again, you may wish to check the answers (use original equation).
EXERCISE. Solve the following quadratic equations. 1. x2 + 5x = 0 2. x2 − 6x = 0 3. x2 − 8x = 0
x(
x = ____
4. x2 + 2x − 8 = 0 5. x2 − 6x − 40 = 0 6. x2 + 5x + 6 = 0
( ) = 0 ( ) = 0 x = ____ x = ____
7. x2 − 49 = 0 8. x2 − 25 = 0 9. x2 − 169 = 0
10. x2 − 4x − 5 = 0 11. x2 − 3x − 10 = 0 12. x2 + 3x − 10 = 0
In the next exercises, remember the first step must be to set the equation equal to zero. 13. x2 + 2x = 24 14. x2 + 5x = 14 15. x2 − 10x = −21 x2 + 2x − 24 = 0 ( )( )=0
[Caution: In #16, when you add −7x to the left side, does it matter where you put it?] 16. x2 + 12 = 7x 17. x2 + 5 = − 6x 18. x2 − 12 = − x
After some practice, you will probably find that it is not necessary to show as many steps in solving problems. The next two examples illustrate the way we usually show the work. EXAMPLE 3. Solve for x: x2 + 21x = 100 EXAMPLE 4. Solve for x: (x−3)(x−4) = 2 Solution: x2 + 21x − 100 = 0 x2 − 7x + 12 = 2 (x + 25)(x − 4) = 0 x2 − 7x + 10 = 0 x = −25 or x = 4 (x − 5)(x − 2) = 0 x = 5 or x = 2
19. x(x + 4) = 5 20. x(x + 5) = 6 21. 3 (2x − 9) = − x2 x2 + 4x − 5 = 0
22. 7x = 18 − x2 23. x (6 − x) = − 40 24. x (5 − x) = − 50
25. 2x2 − 3x = 0 26. 5x2 + 4x = 0 27. x(2x + 7) = −5
28. x(3x + 1) = 2 29. x(5x − 1) = 6 30. x(2x − 9) = 5
In the next exercises, remember to factor the common factor first. 31. 5x2 + 20x + 15 = 0 32. 2x2 − 22x + 48 = 0 5( ) = 0
5¹0 x
+ 0 x = x = Since 5¹0, there are only two solutions, x=____ or _____
33. x3 − 11x2 + 24x = 0 34. 5x3 + 20x2 + 15x = 0 x( ) = 0
x = 0 x − = 0 x − = 0
x = 0 x = x =
There are three solutions:
x=____ , _____, or _____ Compare and contrast #31−32 with #33−34. Notice that a linear equation (with x only) usually has one solution, a quadratic equation (with x2) usually has two solutions, a cubic equation (with x3) usually has three solutions, etc. Notice that the constant factor in #31−32 has no effect on the solutions. However, if the monomial factor has a variable (as in #33−34), it is a solution, and it therefore cannot be ignored.
Does it seem that all quadratic equations have two solutions? Consider the following: 35. x2 − 6x + 9 = 0 36. x2 + 10x + 25 = 0 ( )( ) = 0 x= ____ x= ____ (It is not necessary to write the answer twice!)
37. 4x2 + 20x + 25 = 0 38. 4x2 + 9 = 12x
Although each of the last four exercises is quadratic, each has a single solution that occurrs twice. In higher math courses, we call the solutions to an equation the roots of the equation. When an answer occurs twice, as in these exercises, we call it a double root. It is also possible that the quadratic equation does not factor. These equations can be solved by other methods (see Completing the Square and Quadratic Formula) that will be explained later. Sometimes there is no real solution. For example, x2 + 4 = 0 has no real solution. It does not factor (does it?). Besides, what real number can you square, then add 4 and still end up with zero. If you square a negative number, you get a positive; if you square a positive number, you get a positive number; and if you square 0 you get 0. And in all of these cases, if you add 4 to the result, you certainly do not get 0. Therefore, there is no real solution. EXERCISES. Solve the equations: 39. x3 + 5x2 + 4x = 0 40. x3 − 8x2 + 7x = 0
41. x3 − 9x = 0 42. x3 − 25x = 0
43. 3x2 − 75 = 0 44. 3x3 − 75x = 0
45. x3 − 12x2 + 36x = 0 46. x3 − 9x2 = 0
47. x (x + 2) = 8 48. x (x − 2) = 8
49. (x−3)(x+3) = 8x 50. (x−3)(x−2) = 12
51. x2 = 4(3 − x) 52. x2 = 4(3 + x)
53. 2x2 = 3 + 5x 54. 2x2 = 5x − 3
55. (x − 4)2 = 2x 56. (x − 4)2 = 32 − 2x
57. x(x−4) = −2x + 8 58. 2x (x + 5) = 3x + 15
ANSWERS 2.09 p. 186-192: (NOTE: Answers may be given in any order!) 1. 0, -5; 2. 0, 6; 3. 0, 8; 4. -4, 2; 5. 10, -4; 6. -2, -3; 7. 7, -7; 8. 5, -5; 9. 13, -13; 10. 5,-1; 11. 5,-2; 12. -5,2; 13. -6,4; 14. -7, 2; 15. 3,7; 16. 4,3; 17. -5,-1; 18. -4,3; 19. -5, 1; 20. -6, 1; 21. -9, 3; 22. -9, 2; 23. 10, -4; 24. 10, -5; 25. 0, 3/2; 26. 0, -4/5; 27. -5/2, -1; 28. 2/3, -1; 29. 6/5, -1; 30. -1/2, 5; 31. -3, -1; 32. 8, 3; 33. 0, 8, 3; 34. 0, -3, -1; 35. 3; 36. -5; 37. -5/2; 38. 3/2; 39. 0, -4, -1; 40. 0, 7, 1; 41. 0, 3, -3; 42. 0, 5, -5; 43. 5, -5; 44. 0, 5, -5; 45. 0, 6; 46. 0, 9; 47. -4, 2; 48. 4, -2; 49. 9, -1; 50. 6, -1; 51. -6, 2; 52. 6, -2; 53. -1/2, 3; 54. 3/2, 1; 55. 8, 2; 56. 8, -2; 57. 4, -2; 58. 3/2, -5. Return to main page Math in Living C O L O R !! Return to Basic Algebra page Return to Intermediate Algebra page
|
