 1.2.1: Factor x2  5x  6
 1.2.2: Factor: 2x2  x  3
 1.2.3: The solution set of the equation is
 1.2.4: True or False 4x2 = x
 1.2.5: Complete the square of . Factor the new expression.
 1.2.6: The quantity is called the of a quadratic equation. If it is , the ...
 1.2.7: True or False Quadratic equations always have two real solutions.
 1.2.8: True or False If the discriminant of a quadratic equation is positi...
 1.2.9: In 928, solve each equation by factoring. x2  9x = 0
 1.2.10: In 928, solve each equation by factoring. x2 + 4x = 0
 1.2.11: In 928, solve each equation by factoring. x2  25 = 0
 1.2.12: In 928, solve each equation by factoring. x2  9 = 0
 1.2.13: In 928, solve each equation by factoring. z2 + z  6 = 0
 1.2.14: In 928, solve each equation by factoring. v2 + 7v + 6 = 0
 1.2.15: In 928, solve each equation by factoring. 2x2  5x  3 = 0
 1.2.16: In 928, solve each equation by factoring. 3x2 + 5x + 2 = 0
 1.2.17: In 928, solve each equation by factoring. 3t2  48 = 0
 1.2.18: In 928, solve each equation by factoring. 2y2  50 = 0
 1.2.19: In 928, solve each equation by factoring. x1x  82 + 12 = 0
 1.2.20: In 928, solve each equation by factoring. x1x + 42 = 12
 1.2.21: In 928, solve each equation by factoring. 4x2 + 9 = 12x
 1.2.22: In 928, solve each equation by factoring. 25x2 + 16 = 40x
 1.2.23: In 928, solve each equation by factoring. 61p2  12 = 5p
 1.2.24: In 928, solve each equation by factoring. 212u2  4u2 + 3 = 0
 1.2.25: In 928, solve each equation by factoring. 6x  5 = 6 x
 1.2.26: In 928, solve each equation by factoring. x + 12 x = 7
 1.2.27: In 928, solve each equation by factoring. 41x  22 x  3 + 3 x = 3...
 1.2.28: In 928, solve each equation by factoring. 5 x + 4 = 4 + 3 x  2
 1.2.29: In 2934, solve each equation by the Square Root Method. x = 36 2 = 25
 1.2.30: In 2934, solve each equation by the Square Root Method. x = 4 2 x = 36
 1.2.31: In 2934, solve each equation by the Square Root Method. 1x  122 x = 4
 1.2.32: In 2934, solve each equation by the Square Root Method. 1x + 22 = 9...
 1.2.33: In 2934, solve each equation by the Square Root Method. 12y + 32 = ...
 1.2.34: In 2934, solve each equation by the Square Root Method. 13z  222
 1.2.35: In 3540, solve each equation by completing the square. 2 + 4x = 21
 1.2.36: In 3540, solve each equation by completing the square. .x = 0 2 x ...
 1.2.37: In 3540, solve each equation by completing the square. x2  1 2 x ...
 1.2.38: In 3540, solve each equation by completing the square. x = 0 2 + 2 ...
 1.2.39: In 3540, solve each equation by completing the square. 2 + x  1 2 ...
 1.2.40: In 3540, solve each equation by completing the square. 2x2 3x  3x ...
 1.2.41: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.42: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.43: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.44: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.45: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.46: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.47: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.48: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.49: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.50: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.51: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.52: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.53: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.54: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.55: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.56: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.57: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.58: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.59: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.60: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.61: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.62: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.63: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.64: In 4164, find the real solutions, if any, of each equation. Use the...
 1.2.65: In 6570, find the real solutions, if any, of each equation. Use the...
 1.2.66: In 6570, find the real solutions, if any, of each equation. Use the...
 1.2.67: In 6570, find the real solutions, if any, of each equation. Use the...
 1.2.68: In 6570, find the real solutions, if any, of each equation. Use the...
 1.2.69: In 6570, find the real solutions, if any, of each equation. Use the...
 1.2.70: In 6570, find the real solutions, if any, of each equation. Use the...
 1.2.71: In 7176, use the discriminant to determine whether each quadratic e...
 1.2.72: In 7176, use the discriminant to determine whether each quadratic e...
 1.2.73: In 7176, use the discriminant to determine whether each quadratic e...
 1.2.74: In 7176, use the discriminant to determine whether each quadratic e...
 1.2.75: In 7176, use the discriminant to determine whether each quadratic e...
 1.2.76: In 7176, use the discriminant to determine whether each quadratic e...
 1.2.77: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.78: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.79: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.80: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.81: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.82: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.83: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.84: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.85: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.86: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.87: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.88: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.89: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.90: In 7790, find the real solutions, if any, of each equation. Use any...
 1.2.91: Pythagorean Theorem How many right triangles have a hypotenuse that...
 1.2.92: Pythagorean Theorem How many right triangles have a hypotenuse that...
 1.2.93: Dimensions of a Window The area of the opening of a rectangular win...
 1.2.94: Dimensions of a Window The area of a rectangular window is to be 30...
 1.2.95: Geometry Find the dimensions of a rectangle whose perimeter is 26 m...
 1.2.96: Watering a Field An adjustable water sprinkler that sprays water in...
 1.2.97: Constructing a Box An open box is to be constructed from a square p...
 1.2.98: Constructing a Box Rework if the piece of sheet metal is a rectangl...
 1.2.99: Physics A ball is thrown vertically upward from the top of a buildi...
 1.2.100: Physics An object is propelled vertically upward with an initial ve...
 1.2.101: Reducing the Size of a Candy Bar A jumbo chocolate bar with a recta...
 1.2.102: Reducing the Size of a Candy Bar Rework if the reduction is to be 20%.
 1.2.103: Constructing a Border around a Pool A circular pool measures 10 fee...
 1.2.104: Constructing a Border around a Pool Rework if the depth of the bord...
 1.2.105: Constructing a Border around a Garden A landscaper, who just comple...
 1.2.106: Dimensions of a Patio A contractor orders 8 cubic yards of premixed...
 1.2.107: Comparing TVs The screen size of a television is determined by the ...
 1.2.108: Comparing TVs Refer to 107. Find the screen area of a traditional 5...
 1.2.109: The sum of the consecutive integers is given by the formula How man...
 1.2.110: Geometry If a polygon of n sides has diagonals, how many sides will...
 1.2.111: Show that the sum of the roots of a quadratic equation is
 1.2.112: Show that the product of the roots of a quadratic equation is
 1.2.113: Find k such that the equation has a repeated real solution.
 1.2.114: Find k such that the equation has a repeated real solution.
 1.2.115: Show that the real solutions of the equation are the negatives of t...
 1.2.116: Show that the real solutions of the equation are the reciprocals of...
 1.2.117: Which of the following pairs of equations are equivalent? Explain
 1.2.118: Describe three ways that you might solve a quadratic equation. Stat...
 1.2.119: Explain the benefits of evaluating the discriminant of a quadratic ...
 1.2.120: Create three quadratic equations: one having two distinct solutions...
 1.2.121: The word quadratic seems to imply four (quad), yet a quadratic equa...
Solutions for Chapter 1.2: Quadratic Equations
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 1.2: Quadratic Equations
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters and their solutions. Since 121 problems in chapter 1.2: Quadratic Equations have been answered, more than 9208 students have viewed full stepbystep solutions from this chapter. Chapter 1.2: Quadratic Equations includes 121 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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