 53.1: 4, 7
 53.2: 1 2 , _4 3
 53.3:  _3 5 ,  _1 3
 53.4: x 3  27
 53.5: 4 xy 2  16x
 53.6: 3 x 2 + 8x + 5
 53.7: x 2  11x = 0
 53.8: x 2 + 6x  16 = 0
 53.9: 4x 2  13x = 12
 53.10: x 2  14x = 49
 53.11: x 2 + 9 = 6x
 53.12: x 2  3x =  _9 4
 53.13: " X Y
 53.14: 12 8 4 12 8 16 20 8642
 53.15: 4, 5
 53.16: 6, 8
 53.17: x 2  7x + 6
 53.18: x 2 + 8x  9
 53.19: 3 x 2 + 12x  63
 53.20: 5 x 2  80
 53.21: x 2 + 5x  24 = 0
 53.22: x 2  3x  28 = 0
 53.23: x 2 = 25
 53.24: x 2 = 81
 53.25: x 2 + 3x = 18
 53.26: x 2  4x = 21
 53.27: 2 x 2 + 12x  16 = 0
 53.28: 3 x 2  6x + 9 = 0
 53.29: x 2 + 36 = 12x
 53.30: x 2 + 64 = 16x
 53.31: NUMBER THEORY Find two consecutive even integers with a product of ...
 53.32: PHOTOGRAPHY A rectangular photograph is 8 centimeters wide and 12 c...
 53.33: 3 x 2 = 5x
 53.34: 4 x 2 = 3x
 53.35: 4 x 2 + 7x = 2
 53.36: 4 x 2  17x = 4
 53.37: 4 x 2 + 8x = 3
 53.38: 6 x 2 + 6 = 13x
 53.39: 9x 2 + 30x = 16
 53.40: 1 6x 2  48x = 27
 53.41: Find the roots of x(x + 6)(x  5) = 0.
 53.42: Solve x 3 = 9x by factoring
 53.43: O x y
 53.44: y O 123 4567 x 12 16 4
 53.45:  _2 3 , _3 4
 53.46: _3 2 ,  _4 5
 53.47: DIVING To avoid hitting any rocks below, a cliff diver jumps up and...
 53.48: Rewrite Doyles formula for logs that are 16 feet long.
 53.49: Find the root(s) of the quadratic equation you wrote in Exercise 48...
 53.50: FIND THE ERROR Lina and Kristin are solving x 2 + 2x = 8. Who is co...
 53.51: OPEN ENDED Choose two integers. Then write an equation with those r...
 53.52: CHALLENGE For a quadratic equation of the form (x  p)(x  q) = 0, ...
 53.53: Writing in Math Use the information on page 253 to explain how to s...
 53.54: ACT/SAT Which quadratic equation has roots _1 2 and _1 3 ? A 5 x 2 ...
 53.55: REVIEW What is the solution set for the equation 3(4x + 1) 2 = 48? ...
 53.56: 0 =  x 2  4x + 5
 53.57: 0 = 4 x 2 + 4x + 1
 53.58: 0 = 3 x 2  10x  4
 53.59: Determine whether f(x) = 3 x 2  12x  7 has a maximum or a minimum...
 53.60: CAR MAINTENANCE Vince needs 12 quarts of a 60% antifreeze solution...
 53.61: y O
 53.62: y O x
 53.63: 2x + 4y + 3z = 2x + 3z + 4y
 53.64: 3(6x  7y) = 3(6x) + 3(7y)
 53.65: (3 + 4) + x = 3 + (4 + x)
 53.66: (5x)(3y)(6) = (3y)(6)(5x)
Solutions for Chapter 53: Solving Quadratic Equations by Factoring
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 53: Solving Quadratic Equations by Factoring
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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.)

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.