 1.5.1: Solve each equation in Exercises 114 by factoring. >J  3x  10  0
 1.5.2: Solve each equation in Exercises 114 by factoring. x2  13x + 36  0
 1.5.3: Solve each equation in Exercises 114 by factoring. <'  S.< 15
 1.5.4: Solve each equation in Exercises 114 by factoring. <'  nx  10
 1.5.5: Solve each equation in Exercises 114 by factoring. .6.rl+l lxIO  O
 1.5.6: Solve each equation in Exercises 114 by factoring. 9.r 2 +9x+2  0
 1.5.7: Solve each equation in Exercises 114 by factoring. 1r  2  s
 1.5.8: Solve each equation in Exercises 114 by factoring. 4r  11<   3
 1.5.9: Solve each equation in Exercises 114 by factoring. l<' + 12x  o
 1.5.10: Solve each equation in Exercises 114 by factoring. s.<'  20  o
 1.5.11: Solve each equation in Exercises 114 by factoring. lr(x  3)  5x'...
 1.5.12: Solve each equation in Exercises 114 by factoring. 16.T(x 2)  8x...
 1.5.13: Solve each equation in Exercises 114 by factoring. 7  7x  (1<  ...
 1.5.14: Solve each equation in Exercises 114 by factoring. lOx I  (lx + 1)1
 1.5.15: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.16: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.17: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.18: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.19: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.20: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.21: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.22: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.23: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.24: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.25: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.26: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.27: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.28: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.29: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.30: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.31: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.32: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.33: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.34: Solve the equation in Exercises 1534 by the square root property. ...
 1.5.35: In Exercises 3546, determine the constant that should be added to ...
 1.5.36: In Exercises 3546, determine the constant that should be added to ...
 1.5.37: In Exercises 3546, determine the constant that should be added to ...
 1.5.38: In Exercises 3546, determine the constant that should be added to ...
 1.5.39: In Exercises 3546, determine the constant that should be added to ...
 1.5.40: In Exercises 3546, determine the constant that should be added to ...
 1.5.41: In Exercises 3546, determine the constant that should be added to ...
 1.5.42: In Exercises 3546, determine the constant that should be added to ...
 1.5.43: In Exercises 3546, determine the constant that should be added to ...
 1.5.44: In Exercises 3546, determine the constant that should be added to ...
 1.5.45: In Exercises 3546, determine the constant that should be added to ...
 1.5.46: In Exercises 3546, determine the constant that should be added to ...
 1.5.47: Solve each equation in Exercises 4764 by completing the square. <1...
 1.5.48: Solve each equation in Exercises 4764 by completing the square. <'...
 1.5.49: Solve each equation in Exercises 4764 by completing the square. r...
 1.5.50: Solve each equation in Exercises 4764 by completing the square. r ...
 1.5.51: Solve each equation in Exercises 4764 by completing the square. x'...
 1.5.52: Solve each equation in Exercises 4764 by completing the square. .r...
 1.5.53: Solve each equation in Exercises 4764 by completing the square. .r...
 1.5.54: Solve each equation in Exercises 4764 by completing the square. x1...
 1.5.55: Solve each equation in Exercises 4764 by completing the square. x'...
 1.5.56: Solve each equation in Exercises 4764 by completing the square. x'...
 1.5.57: Solve each equation in Exercises 4764 by completing the square. .r...
 1.5.58: Solve each equation in Exercises 4764 by completing the square. r...
 1.5.59: Solve each equation in Exercises 4764 by completing the square. lx...
 1.5.60: Solve each equation in Exercises 4764 by completing the square. 1r...
 1.5.61: Solve each equation in Exercises 4764 by completing the square. l...
 1.5.62: Solve each equation in Exercises 4764 by completing the square. 2x...
 1.5.63: Solve each equation in Exercises 4764 by completing the square. 1r...
 1.5.64: Solve each equation in Exercises 4764 by completing the square. 1<...
 1.5.65: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.66: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.67: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.68: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.69: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.70: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.71: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.72: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.73: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.74: Solve each equation in Exercises 6574 using the quadratic formula....
 1.5.75: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.76: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.77: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.78: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.79: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.80: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.81: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.82: In Exercises 7582. compute the discriminant. Then determine the nu...
 1.5.83: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.84: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.85: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.86: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.87: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.88: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.89: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.90: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.91: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.92: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.93: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.94: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.95: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.96: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.97: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.98: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.99: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.100: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.101: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.102: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.103: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.104: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.105: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.106: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.107: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.108: Solve each equation in Exercises 83108 by the method of your choic...
 1.5.109: In Exercises 109114, find tire ximercept(s) of tire graph of each...
 1.5.110: In Exercises 109114, find tire ximercept(s) of tire graph of each...
 1.5.111: In Exercises 109114, find tire ximercept(s) of tire graph of each...
 1.5.112: In Exercises 109114, find tire ximercept(s) of tire graph of each...
 1.5.113: In Exercises 109114, find tire ximercept(s) of tire graph of each...
 1.5.114: In Exercises 109114, find tire ximercept(s) of tire graph of each...
 1.5.115: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.116: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.117: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.118: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.119: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.120: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.121: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.122: In Exercises 115122 find all values of x satisfying the given cond...
 1.5.123: In Exercises 123124, list all numbers thai must be excluded from t...
 1.5.124: In Exercises 123124, list all numbers thai must be excluded from t...
 1.5.125: When the sum of 6 and twice a positive number is subtracted from th...
 1.5.126: When the sum of 1 and twice a negative numbe is subtracted from twi...
 1.5.127: In Exercises 127130. solve each equation by the method of your cho...
 1.5.128: In Exercises 127130. solve each equation by the method of your cho...
 1.5.129: In Exercises 127130. solve each equation by the method of your cho...
 1.5.130: In Exercises 127130. solve each equation by the method of your cho...
 1.5.131: In a roundrobin chess toumament, e.ach player is paired with every...
 1.5.132: In a roundrobin chess toumament, e.ach player is paired with every...
 1.5.133: The graph of the fomwla in Exercises 131132 is shown. Use the grap...
 1.5.134: The graph of the fomwla in Exercises 131132 is shown. Use the grap...
 1.5.135: A driver's age has something to do with his or her chanc.e of getti...
 1.5.136: A driver's age has something to do with his or her chanc.e of getti...
 1.5.137: Throwing events in track and field include the shot put, the discus...
 1.5.138: Throwing events in track and field include the shot put, the discus...
 1.5.139: If you have not yet done so, read the Blitzer Bonus on page 159. In...
 1.5.140: Use the Pythagorean Theorem and the squatY root pro~rry, to solve E...
 1.5.141: Use the Pythagorean Theorem and the squatY root pro~rry, to solve E...
 1.5.142: Use the Pythagorean Theorem and the squatY root pro~rry, to solve E...
 1.5.143: Use the Pythagorean Theorem and the squatY root pro~rry, to solve E...
 1.5.144: An isosceles righl tria ngle hus lcgslh:tl nrc the snmc length and ...
 1.5.145: The length of a rectangulor sign is 3 feet longer than the wdth. If...
 1.5.146: A rectangular parking lot has a length that is 3 yards greater than...
 1.5.147: Each side of a square is lengthened by 3 inches. The area of this n...
 1.5.148: Each side of a square is lengthened by 2 inches. The area of this n...
 1.5.149: A pool measuring 10 meters by 20 meters is surrounded by a path of ...
 1.5.150: A vacant rectangular lot is being turned into a community vegetable...
 1.5.151: A machine produces open boxes using square sheets or metaL "llte fi...
 1.5.152: A machine produces open boxes using square sheets of metal. The mac...
 1.5.153: A rain gutter is made fTom sheets or aluminum lhat arc 20 inches "i...
 1.5.154: A piece of wire is 8 inches long. The wire is cut into two pieces a...
 1.5.155: What is a quadratic equation?
 1.5.156: Explain how to solve x1 + 6x + 8  0 using factoring and the zerop...
 1.5.157: Explain how to solve x2 + 6x + 8  0 by completing the square.
 1.5.158: Explain how to .solve x1 + 6x + 8  0 using the quadratic formula.
 1.5.159: How is the quadratic fomula derived?
 1.5.160: What is the discriminant and what information does it provide about...
 1.5.161: If you are given a quadratic equation, how do you determine which m...
 1.5.162: Describe the relationship between the real solutions of ax1 + bx + ...
 1.5.163: If a quadratic equation has imaginary solutions.., how is this show...
 1.5.164: Use a graphing utility and xintercepts to verify any of the real s...
 1.5.165: Use a graphing utility to graph y  ax2 + bx + c related to any fi\...
 1.5.166: In Exercises 166169, determine whether each statement makes sense ...
 1.5.167: In Exercises 166169, determine whether each statement makes sense ...
 1.5.168: In Exercises 166169, determine whether each statement makes sense ...
 1.5.169: In Exercises 166169, determine whether each statement makes sense ...
 1.5.170: In Exercises 170173. determine whether each stalement is true or f...
 1.5.171: In Exercises 170173. determine whether each stalement is true or f...
 1.5.172: In Exercises 170173. determine whether each stalement is true or f...
 1.5.173: In Exercises 170173. determine whether each stalement is true or f...
 1.5.174: Write a q uadratic equation in general form whose solution set is (...
 1.5.175: Solve for t.s   16i1 + .
 1.5.176: A rectangular swimming pool is 12 meters long and 8 meters wide. A ...
 1.5.177: Exercises 177179 will help you prepare for the material covered in...
 1.5.178: Exercises 177179 will help you prepare for the material covered in...
 1.5.179: Exercises 177179 will help you prepare for the material covered in...
Solutions for Chapter 1.5: Quadratic Equations
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 1.5: Quadratic Equations
Get Full SolutionsChapter 1.5: Quadratic Equations includes 179 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 179 problems in chapter 1.5: Quadratic Equations have been answered, more than 39472 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 6. College Algebra was written by and is associated to the ISBN: 9780321782281.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).