 5.6.1: Find the exact solutions by using the Quadratic Formula.
 5.6.2: Find the exact solutions by using the Quadratic Formula.
 5.6.3: Find the exact solutions by using the Quadratic Formula.
 5.6.4: Find the exact solutions by using the Quadratic Formula.
 5.6.5: Find the exact solutions by using the Quadratic Formula.
 5.6.6: Find the exact solutions by using the Quadratic Formula.
 5.6.7: Find the exact solutions by using the Quadratic Formula.
 5.6.8: Find the exact solutions by using the Quadratic Formula.
 5.6.9: When will the object be at a height of 50 feet?
 5.6.10: Will the object ever reach a height of 120 feet? Explain your reaso...
 5.6.11: Complete parts a and b for each quadratic equation. a. Find the val...
 5.6.12: Complete parts a and b for each quadratic equation. a. Find the val...
 5.6.13: Complete parts a and b for each quadratic equation. a. Find the val...
 5.6.14: Complete parts a and b for each quadratic equation. a. Find the val...
 5.6.15: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.16: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.17: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.18: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.19: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.20: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.21: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.22: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.23: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.24: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.25: Solve each equation by using the method of your choice. Find exact ...
 5.6.26: Solve each equation by using the method of your choice. Find exact ...
 5.6.27: Solve each equation by using the method of your choice. Find exact ...
 5.6.28: Solve each equation by using the method of your choice. Find exact ...
 5.6.29: Solve each equation by using the method of your choice. Find exact ...
 5.6.30: Solve each equation by using the method of your choice. Find exact ...
 5.6.31: Determine a domain and range for which this function makes sense
 5.6.32: According to this model, in what year did the average salary first ...
 5.6.33: Highway safety engineers can use the formula d = 0.05s 2 + 1.1s to ...
 5.6.34: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.35: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.36: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.37: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.38: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.39: Complete parts ac for each quadratic equation. a. Find the value of...
 5.6.40: Solve each equation by using the method of your choice. Find exact ...
 5.6.41: Solve each equation by using the method of your choice. Find exact ...
 5.6.42: Solve each equation by using the method of your choice. Find exact ...
 5.6.43: Solve each equation by using the method of your choice. Find exact ...
 5.6.44: Solve each equation by using the method of your choice. Find exact ...
 5.6.45: Solve each equation by using the method of your choice. Find exact ...
 5.6.46: Calculate the value of the discriminant
 5.6.47: What does the discriminant tell you about the supporting cables of ...
 5.6.48: Civil engineers are designing a section of road that is going to di...
 5.6.49: Graph a quadratic equation that has a a. positive discriminant. b. ...
 5.6.50: Explain why the roots of a quadratic equation are complex if the va...
 5.6.51: Find the exact solutions of 2ix2  3ix  5i = 0 by using the Quadra...
 5.6.52: Given the equation x2 + 3x  4 = 0, a. Find the exact solutions by ...
 5.6.53: Use the information on page 276 to explain how a divers height abov...
 5.6.54: If 2x 2  5x  9 = 0, then x could be approximately equal to which ...
 5.6.55: What are the xintercepts of the graph of y = 2x2  5x + 12? F  _...
 5.6.56: Solve each equation by using the Square Root Property. (
 5.6.57: Solve each equation by using the Square Root Property. (
 5.6.58: Solve each equation by using the Square Root Property. (
 5.6.59: Simplify.
 5.6.60: Simplify.
 5.6.61: Simplify.
 5.6.62: Solve each system of inequalities.
 5.6.63: Solve each system of inequalities.
 5.6.64: Write the slopeintercept form of the equation of the line with eac...
 5.6.65: Write the slopeintercept form of the equation of the line with eac...
 5.6.66: Write the slopeintercept form of the equation of the line with eac...
 5.6.67: State whether each trinomial is a perfect square. If so, factor it.
 5.6.68: State whether each trinomial is a perfect square. If so, factor it.
 5.6.69: State whether each trinomial is a perfect square. If so, factor it.
 5.6.70: State whether each trinomial is a perfect square. If so, factor it.
 5.6.71: State whether each trinomial is a perfect square. If so, factor it.
 5.6.72: State whether each trinomial is a perfect square. If so, factor it.
Solutions for Chapter 5.6: The Quadratic Formula and the Discriminant
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 5.6: The Quadratic Formula and the Discriminant
Get Full SolutionsThis textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Since 72 problems in chapter 5.6: The Quadratic Formula and the Discriminant have been answered, more than 44240 students have viewed full stepbystep solutions from this chapter. Chapter 5.6: The Quadratic Formula and the Discriminant includes 72 full stepbystep solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. This expansive textbook survival guide covers the following chapters and their solutions.

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).

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

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Iterative method.
A sequence of steps intended to approach the desired solution.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Outer product uv T
= column times row = rank one matrix.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Solvable system Ax = b.
The right side b is in the column space of A.