 8.2.1: The solutions of a quadratic equation in standard form ax2 + bx + c...
 8.2.2: In order to solve 2x2 + 9x  5 = 0 by the quadratic formula, we use...
 8.2.3: In order to solve x2 = 4x + 1 by the quadratic formula, we use a = ...
 8.2.4: x = (4) { 2(4) 2  4(1)(2) 2(1) simplifies to x = ______________.
 8.2.5: x = 4 { 242  4 # 2 # 5 2 # 2 simplifies to x = ______________.
 8.2.6: The discriminant of ax2 + bx + c = 0 is defined by ______________.
 8.2.7: If the discriminant of ax2 + bx + c = 0 is negative, the quadratic ...
 8.2.8: If the discriminant of ax2 + bx + c = 0 is positive, the quadratic ...
 8.2.9: The most efficient technique for solving (2x + 7) 2 = 25 is by usin...
 8.2.10: The most efficient technique for solving x2 + 5x  10 = 0 is by usi...
 8.2.11: The most efficient technique for solving x2 + 8x + 15 = 0 is by usi...
 8.2.12: True or false: An equation with the solution set 52, 56 is (x + 2)(...
 8.2.13: In Exercises 118, solve each equation using the quadratic formula. ...
 8.2.14: In Exercises 118, solve each equation using the quadratic formula. ...
 8.2.15: In Exercises 118, solve each equation using the quadratic formula. ...
 8.2.16: In Exercises 118, solve each equation using the quadratic formula. ...
 8.2.17: In Exercises 118, solve each equation using the quadratic formula. ...
 8.2.18: In Exercises 118, solve each equation using the quadratic formula. ...
 8.2.19: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.20: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.21: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.22: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.23: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.24: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.25: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.26: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.27: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.28: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.29: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.30: In Exercises 1930, compute the discriminant. Then determine the num...
 8.2.31: In Exercises 3150, solve each equation by the method of your choice...
 8.2.32: In Exercises 3150, solve each equation by the method of your choice...
 8.2.33: In Exercises 3150, solve each equation by the method of your choice...
 8.2.34: In Exercises 3150, solve each equation by the method of your choice...
 8.2.35: In Exercises 3150, solve each equation by the method of your choice...
 8.2.36: In Exercises 3150, solve each equation by the method of your choice...
 8.2.37: In Exercises 3150, solve each equation by the method of your choice...
 8.2.38: In Exercises 3150, solve each equation by the method of your choice...
 8.2.39: In Exercises 3150, solve each equation by the method of your choice...
 8.2.40: In Exercises 3150, solve each equation by the method of your choice...
 8.2.41: In Exercises 3150, solve each equation by the method of your choice...
 8.2.42: In Exercises 3150, solve each equation by the method of your choice...
 8.2.43: In Exercises 3150, solve each equation by the method of your choice...
 8.2.44: In Exercises 3150, solve each equation by the method of your choice...
 8.2.45: In Exercises 3150, solve each equation by the method of your choice...
 8.2.46: In Exercises 3150, solve each equation by the method of your choice...
 8.2.47: In Exercises 3150, solve each equation by the method of your choice...
 8.2.48: In Exercises 3150, solve each equation by the method of your choice...
 8.2.49: In Exercises 3150, solve each equation by the method of your choice...
 8.2.50: In Exercises 3150, solve each equation by the method of your choice...
 8.2.51: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.52: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.53: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.54: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.55: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.56: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.57: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.58: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.59: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.60: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.61: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.62: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.63: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.64: In Exercises 5164, write a quadratic equation in standard form with...
 8.2.65: Exercises 6568 describe quadratic equations. Match each description...
 8.2.66: Exercises 6568 describe quadratic equations. Match each description...
 8.2.67: Exercises 6568 describe quadratic equations. Match each description...
 8.2.68: Exercises 6568 describe quadratic equations. Match each description...
 8.2.69: When the sum of 6 and twice a positive number is subtracted from th...
 8.2.70: When the sum of 1 and twice a negative number is subtracted from tw...
 8.2.71: In Exercises 7176, solve each equation by the method of your choice...
 8.2.72: In Exercises 7176, solve each equation by the method of your choice...
 8.2.73: In Exercises 7176, solve each equation by the method of your choice...
 8.2.74: In Exercises 7176, solve each equation by the method of your choice...
 8.2.75: In Exercises 7176, solve each equation by the method of your choice...
 8.2.76: In Exercises 7176, solve each equation by the method of your choice...
 8.2.77: Use the function to solve Exercises 7778.What age groups are expect...
 8.2.78: Use the function to solve Exercises 7778.What age groups are expect...
 8.2.79: In Exercises 7980, an athlete whose event is the shot put releases ...
 8.2.80: In Exercises 7980, an athlete whose event is the shot put releases ...
 8.2.81: The length of a rectangle is 4 meters longer than the width. If the...
 8.2.82: The length of a rectangle exceeds twice its width by 3 inches. If t...
 8.2.83: The longer leg of a right triangle exceeds the shorter leg by 1 inc...
 8.2.84: The hypotenuse of a right triangle is 6 feet long. One leg is 2 fee...
 8.2.85: A rain gutter is made from sheets of aluminum that are 20 inches wi...
 8.2.86: A piece of wire is 8 inches long. The wire is cut into two pieces a...
 8.2.87: Working together, two people can mow a large lawn in 4 hours. One p...
 8.2.88: A pool has an inlet pipe to fill it and an outlet pipe to empty it....
 8.2.89: What is the quadratic formula and why is it useful?
 8.2.90: Without going into specific details for every step, describe how th...
 8.2.91: Explain how to solve x2 + 6x + 8 = 0 using the quadratic formula.
 8.2.92: If a quadratic equation has imaginary solutions, how is this shown ...
 8.2.93: What is the discriminant and what information does it provide about...
 8.2.94: If you are given a quadratic equation, how do you determine which m...
 8.2.95: Explain how to write a quadratic equation from its solution set. Gi...
 8.2.96: Use a graphing utility to graph the quadratic function related to a...
 8.2.97: Reread Exercise 85. The crosssectional area of the gutter is given...
 8.2.98: Make Sense? In Exercises 98101, determine whether each statement ma...
 8.2.99: Make Sense? In Exercises 98101, determine whether each statement ma...
 8.2.100: Make Sense? In Exercises 98101, determine whether each statement ma...
 8.2.101: Make Sense? In Exercises 98101, determine whether each statement ma...
 8.2.102: In Exercises 102105, determine whether each statement is true or fa...
 8.2.103: In Exercises 102105, determine whether each statement is true or fa...
 8.2.104: In Exercises 102105, determine whether each statement is true or fa...
 8.2.105: In Exercises 102105, determine whether each statement is true or fa...
 8.2.106: Solve for t: s = 16t 2 + v0t.
 8.2.107: A rectangular swimming pool is 12 meters long and 8 meters wide. A ...
 8.2.108: The area of the shaded green region outside the rectangle and insid...
 8.2.109: Solve: 5x + 2 = 4  3x . (Section 4.3, Example 3)
 8.2.110: Solve: 22x  5  2x  3 = 1. (Section 7.6, Example 4)
 8.2.111: Rationalize the denominator: 5 23 + x . (Section 7.5, Example 5)
 8.2.112: Exercises 112114 will help you prepare for the material covered in ...
 8.2.113: Exercises 112114 will help you prepare for the material covered in ...
 8.2.114: Exercises 112114 will help you prepare for the material covered in ...
Solutions for Chapter 8.2: The Quadratic Formula
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 8.2: The Quadratic Formula
Get Full SolutionsSince 114 problems in chapter 8.2: The Quadratic Formula have been answered, more than 89652 students have viewed full stepbystep solutions from this chapter. Chapter 8.2: The Quadratic Formula includes 114 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

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.

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).

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 variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).