- 9.1.1: Use a graph and table to approximate solutions for each equation, t...
- 9.1.2: Classify each number by specifying all of the number sets of which ...
- 9.1.3: Use a symbolic method to solve each equation. Show each solution ex...
- 9.1.4: Sketch the graph of a quadratic function with a. One x-intercept. b...
- 9.1.5: A baseball is dropped from the top of a very tall building. The bal...
- 9.1.6: APPLICATION A flare is fired into the air from the ground. It reach...
- 9.1.7: The path of a ball in flight is given by p(x) = 0.23(x 3.4)2 + 4.2,...
- 9.1.8: Solve the equation 4 = 2(x 3)2 + 4 using a. A graph. b. A table. c....
- 9.1.9: APPLICATION The graph at right shows the graph of h(t) = 4.9t2 + 50...
- 9.1.10: Mini-Investigation In each equation the variable x represents time ...
- 9.1.11: Show a step-by-step symbolic solution of the inequality 3x + 4 > 16.
- 9.1.12: The solid line in the graph passes through (0, 6) and (6, 1). Write...
Solutions for Chapter 9.1: Solving Quadratic Equations
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Column space C (A) =
space of all combinations of the columns of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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.
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 •
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.