 A.7.1: The square of a complex number is sometimes negative
 A.7.2: 12 + i2 12  i2 =
 A.7.3: In the complex number system, a quadratic equation has four solutions.
 A.7.4: In the complex number 5 + 2i, the number 5 is called the part; the ...
 A.7.5: The equation x2 = 4 has the solution set
 A.7.6: The conjugate of 2 + 5i is 2  5i.
 A.7.7: All real numbers are complex numbers.
 A.7.8: If 2  3i is a solution of a quadratic equation with real coefficie...
 A.7.9: In 9 46, write each expression in the standard form a + bi.
 A.7.10: In 9 46, write each expression in the standard form a + bi.
 A.7.11: In 9 46, write each expression in the standard form a + bi.
 A.7.12: In 9 46, write each expression in the standard form a + bi.
 A.7.13: In 9 46, write each expression in the standard form a + bi.
 A.7.14: In 9 46, write each expression in the standard form a + bi.
 A.7.15: In 9 46, write each expression in the standard form a + bi.
 A.7.16: In 9 46, write each expression in the standard form a + bi.
 A.7.17: In 9 46, write each expression in the standard form a + bi.
 A.7.18: In 9 46, write each expression in the standard form a + bi.
 A.7.19: In 9 46, write each expression in the standard form a + bi.
 A.7.20: In 9 46, write each expression in the standard form a + bi.
 A.7.21: In 9 46, write each expression in the standard form a + bi.
 A.7.22: In 9 46, write each expression in the standard form a + bi.
 A.7.23: In 9 46, write each expression in the standard form a + bi.
 A.7.24: In 9 46, write each expression in the standard form a + bi.
 A.7.25: In 9 46, write each expression in the standard form a + bi.
 A.7.26: In 9 46, write each expression in the standard form a + bi.
 A.7.27: In 9 46, write each expression in the standard form a + bi.
 A.7.28: In 9 46, write each expression in the standard form a + bi.
 A.7.29: In 9 46, write each expression in the standard form a + bi.
 A.7.30: In 9 46, write each expression in the standard form a + bi.
 A.7.31: In 9 46, write each expression in the standard form a + bi.
 A.7.32: In 9 46, write each expression in the standard form a + bi.
 A.7.33: In 9 46, write each expression in the standard form a + bi.
 A.7.34: In 9 46, write each expression in the standard form a + bi.
 A.7.35: In 9 46, write each expression in the standard form a + bi.
 A.7.36: In 9 46, write each expression in the standard form a + bi.
 A.7.37: In 9 46, write each expression in the standard form a + bi.
 A.7.38: In 9 46, write each expression in the standard form a + bi.
 A.7.39: In 9 46, write each expression in the standard form a + bi.
 A.7.40: In 9 46, write each expression in the standard form a + bi.
 A.7.41: In 9 46, write each expression in the standard form a + bi.
 A.7.42: In 9 46, write each expression in the standard form a + bi.
 A.7.43: In 9 46, write each expression in the standard form a + bi.
 A.7.44: In 9 46, write each expression in the standard form a + bi.
 A.7.45: In 9 46, write each expression in the standard form a + bi.
 A.7.46: In 9 46, write each expression in the standard form a + bi.
 A.7.47: In 47 52, perform the indicated operations and express your answer ...
 A.7.48: In 47 52, perform the indicated operations and express your answer ...
 A.7.49: In 47 52, perform the indicated operations and express your answer ...
 A.7.50: In 47 52, perform the indicated operations and express your answer ...
 A.7.51: In 47 52, perform the indicated operations and express your answer ...
 A.7.52: In 47 52, perform the indicated operations and express your answer ...
 A.7.53: In 53 72, solve each equation in the complex number system.
 A.7.54: In 53 72, solve each equation in the complex number system.
 A.7.55: In 53 72, solve each equation in the complex number system.
 A.7.56: In 53 72, solve each equation in the complex number system.
 A.7.57: In 53 72, solve each equation in the complex number system.
 A.7.58: In 53 72, solve each equation in the complex number system.
 A.7.59: In 53 72, solve each equation in the complex number system.
 A.7.60: In 53 72, solve each equation in the complex number system.
 A.7.61: In 53 72, solve each equation in the complex number system.
 A.7.62: In 53 72, solve each equation in the complex number system.
 A.7.63: In 53 72, solve each equation in the complex number system.
 A.7.64: In 53 72, solve each equation in the complex number system.
 A.7.65: In 53 72, solve each equation in the complex number system.
 A.7.66: In 53 72, solve each equation in the complex number system.
 A.7.67: In 53 72, solve each equation in the complex number system.
 A.7.68: In 53 72, solve each equation in the complex number system.
 A.7.69: In 53 72, solve each equation in the complex number system.
 A.7.70: In 53 72, solve each equation in the complex number system.
 A.7.71: In 53 72, solve each equation in the complex number system.
 A.7.72: In 53 72, solve each equation in the complex number system.
 A.7.73: In 73 78, without solving, determine the character of the solutions...
 A.7.74: In 73 78, without solving, determine the character of the solutions...
 A.7.75: In 73 78, without solving, determine the character of the solutions...
 A.7.76: In 73 78, without solving, determine the character of the solutions...
 A.7.77: In 73 78, without solving, determine the character of the solutions...
 A.7.78: In 73 78, without solving, determine the character of the solutions...
 A.7.79: 2 + 3i is a solution of a quadratic equation with real coefficients...
 A.7.80: 4  i is a solution of a quadratic equation with real coefficients....
 A.7.81: In 81 84, z = 3  4i and w = 8 + 3i. Write each expression in the s...
 A.7.82: In 81 84, z = 3  4i and w = 8 + 3i. Write each expression in the s...
 A.7.83: In 81 84, z = 3  4i and w = 8 + 3i. Write each expression in the s...
 A.7.84: In 81 84, z = 3  4i and w = 8 + 3i. Write each expression in the s...
 A.7.85: The impedance Z, in ohms, of a circuit element is defined as the ra...
 A.7.86: In an ac circuit with two parallel pathways, the total impedance Z,...
 A.7.87: Use z = a + bi to show that z + z = 2a and z  z = 2bi.
 A.7.88: Use z = a + bi to show that z = z
 A.7.89: Use z = a + bi and w = c + di to show that z + w = z + w
 A.7.90: Use z = a + bi and w = c + di to show that z # w = z # w.
 A.7.91: Explain to a friend how you would add two complex numbers and how y...
 A.7.92: Write a brief paragraph that compares the method used to rationaliz...
 A.7.93: Use an Internet search engine to investigate the origins of complex...
 A.7.94: A student multiplied 29 and 29 as follows:29 # 29 = 2(9)(9) =...
Solutions for Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System
Get Full SolutionsSince 94 problems in chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System have been answered, more than 76709 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System includes 94 full stepbystep solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·