 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 xintercepts are shown. Evaluat...
 9.7.7: Match each quadratic equation with its graph. Then explain how to u...
 9.7.8: MiniInvestigation 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
ISBN: 9781559537636
Solutions for Chapter 9.7: The Quadratic Formula
Get Full SolutionsChapter 9.7: The Quadratic Formula includes 13 full stepbystep solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Since 13 problems in chapter 9.7: The Quadratic Formula have been answered, more than 8410 students have viewed full stepbystep solutions from this chapter.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Companion 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 nl  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. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
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 Fn1c 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.

Lucas numbers
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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

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.