- 9.7.1: Without using a calculator, evaluate the expression b2 4ac for the ...
- 9.7.2: Rewrite each quadratic equation in general form if necessary. For e...
- 9.7.3: Solve each quadratic equation. Which equation can you solve readily...
- 9.7.4: Graph the equation y = x2 + 3x + 5. Use the graph and the quadratic...
- 9.7.5: The equation h = 4.9t2 + 6.2t + 1.9 models the height of a soccer b...
- 9.7.6: Find an equation whose solutions as x-intercepts are shown. Evaluat...
- 9.7.7: Match each quadratic equation with its graph. Then explain how to u...
- 9.7.8: Mini-Investigation The quadratic formula gives two roots of an equa...
- 9.7.9: The equation h = 4.9t2 + 17t + 2.2 models the height of a stone thr...
- 9.7.10: APPLICATION A shopkeeper is redesigning the rectangular sign on her...
- 9.7.11: Algebraically find the intersection points, if any, of the graphs o...
- 9.7.12: Reduce each rational expression by factoring, then canceling common...
- 9.7.13: On your graph paper, sketch graphs of these equations. Then use you...
Solutions for Chapter 9.7: The Quadratic Formula
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
peA) = det(A - AI) has peA) = zero matrix.
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).
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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 •
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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).
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.