 56.1: 8x 2 + 18x  5 = 0
 56.2: x 2 + 8x = 0
 56.3: 4x 2 + 4x + 1 = 0
 56.4: x 2 + 6x + 9 = 0
 56.5: 2x 2  4x + 1 = 0
 56.6: x 2  2x  2 = 0
 56.7: x 2 + 3x + 8 = 5
 56.8: 4x 2 + 20x + 25 = 2
 56.9: When will the object be at a height of 50 feet?
 56.10: Will the object ever reach a height of 120 feet? Explain your reaso...
 56.11: 8x 2 + 18x  5 = 0
 56.12: 4x 2 + 4x + 1 = 0
 56.13: 2x 2  4x + 1 = 0
 56.14: x 2 + 3x + 8 = 5
 56.15: 12x 2 + 5x + 2 = 0
 56.16: 3x 2  5x + 2 = 0
 56.17: 9x 2  6x  4 = 5
 56.18: 25 + 4x 2 = 20x
 56.19: x 2 + 3x  3 = 0
 56.20: x 2  16x + 4 = 0
 56.21: x 2 + 4x + 3 = 4
 56.22: 2x  5 = x 2
 56.23: x 2  2x + 5 = 0
 56.24: x 2  x + 6 = 0
 56.25: x 2  30x  64 = 0
 56.26: 7x 2 + 3 = 0
 56.27: x 2  4x + 7 = 0
 56.28: 2x 2 + 6x  3 = 0
 56.29: 4x 2  8 = 0
 56.30: 4x 2 + 81 = 36x
 56.31: Determine a domain and range for which this function makes sense.
 56.32: According to this model, in what year did the average salary first ...
 56.33: HIGHWAY SAFETY Highway safety engineers can use the formula d = 0.0...
 56.34: x 2 + 6x = 0
 56.35: 4x 2 + 7 = 9x
 56.36: 3x + 6 = 6x 2
 56.37: 3 4 x 2  _1 3 x  1 = 0
 56.38: 0.4x 2 + x  0.3 = 0
 56.39: 0.2x 2 + 0.1x + 0.7 = 0
 56.40: 4(x + 3) 2 = 28
 56.41: 3x 2  10x = 7
 56.42: x 2 + 9 = 8x
 56.43: 10x 2 + 3x = 0
 56.44: 2x 2  12x + 7 = 5
 56.45: 21 = (x  2) 2 + 5
 56.46: Calculate the value of the discriminant.
 56.47: What does the discriminant tell you about the supporting cables of ...
 56.48: ENGINEERING Civil engineers are designing a section of road that is...
 56.49: OPEN ENDED Graph a quadratic equation that has a a. positive discri...
 56.50: REASONING Explain why the roots of a quadratic equation are complex...
 56.51: CHALLENGE Find the exact solutions of 2ix2  3ix  5i = 0 by using ...
 56.52: REASONING Given the equation x2 + 3x  4 = 0, a. Find the exact sol...
 56.53: Writing in Math Use the information on page 276 to explain how a di...
 56.54: ACT/SAT If 2x 2  5x  9 = 0, then x could be approximately equal t...
 56.55: REVIEW What are the xintercepts of the graph of y = 2x2  5x + 12...
 56.56: x 2 + 18x + 81 = 25
 56.57: x 2  8x + 16 = 7
 56.58: 4x 2  4x + 1 = 8
 56.59: 2i 3 + i
 56.60: _4 5
 56.61: 1 + i 3  2i
 56.62: x + y 9 y  x 4
 56.63: x 1 x  y 3 y 1 y  x 4 y x
 56.64: y O x
 56.65: y O x
 56.66: PHOTOGRAPHY Desiree works in a photography studio and makes a commi...
 56.67: x 2  5x  10
 56.68: x 2  14x + 49
 56.69: 4x 2 + 12x + 9
 56.70: 25x 2 + 20x + 4
 56.71: 9x 2  12x + 16
 56.72: 36x 2  60x + 25
Solutions for Chapter 56: The Quadratic Formula and the Discriminant
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 56: The Quadratic Formula and the Discriminant
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 56: The Quadratic Formula and the Discriminant includes 72 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 72 problems in chapter 56: The Quadratic Formula and the Discriminant have been answered, more than 53728 students have viewed full stepbystep solutions from this chapter. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.