 8.1.1: The square root property states that if u2 = d, then u = __________...
 8.1.2: If x2 = 7, then x = ______________.
 8.1.3: If x2 = 11 2 , then x = ______________. Rationalizing denominators,...
 8.1.4: If x2 = 9, then x = ______________.
 8.1.5: To complete the square on x2 + 10x, add ______________.
 8.1.6: To complete the square on x2  3x, add ______________.
 8.1.7: To complete the square on x2  4 5 x, add ______________.
 8.1.8: To solve x2 + 6x = 7 by completing the square, add ______________ t...
 8.1.9: To solve x2  2 3 x = 4 9 by completing the square, add ___________...
 8.1.10: In Exercises 122, solve each equation by the square root property. ...
 8.1.11: In Exercises 122, solve each equation by the square root property. ...
 8.1.12: In Exercises 122, solve each equation by the square root property. ...
 8.1.13: In Exercises 122, solve each equation by the square root property. ...
 8.1.14: In Exercises 122, solve each equation by the square root property. ...
 8.1.15: In Exercises 122, solve each equation by the square root property. ...
 8.1.16: In Exercises 122, solve each equation by the square root property. ...
 8.1.17: In Exercises 122, solve each equation by the square root property. ...
 8.1.18: In Exercises 122, solve each equation by the square root property. ...
 8.1.19: In Exercises 122, solve each equation by the square root property. ...
 8.1.20: In Exercises 122, solve each equation by the square root property. ...
 8.1.21: In Exercises 122, solve each equation by the square root property. ...
 8.1.22: In Exercises 122, solve each equation by the square root property. ...
 8.1.23: In Exercises 2334, determine the constant that should be added to t...
 8.1.24: In Exercises 2334, determine the constant that should be added to t...
 8.1.25: In Exercises 2334, determine the constant that should be added to t...
 8.1.26: In Exercises 2334, determine the constant that should be added to t...
 8.1.27: In Exercises 2334, determine the constant that should be added to t...
 8.1.28: In Exercises 2334, determine the constant that should be added to t...
 8.1.29: In Exercises 2334, determine the constant that should be added to t...
 8.1.30: In Exercises 2334, determine the constant that should be added to t...
 8.1.31: In Exercises 2334, determine the constant that should be added to t...
 8.1.32: In Exercises 2334, determine the constant that should be added to t...
 8.1.33: In Exercises 2334, determine the constant that should be added to t...
 8.1.34: In Exercises 2334, determine the constant that should be added to t...
 8.1.35: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.36: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.37: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.38: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.39: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.40: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.41: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.42: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.43: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.44: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.45: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.46: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.47: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.48: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.49: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.50: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.51: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.52: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.53: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.54: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.55: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.56: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.57: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.58: In Exercises 3558, solve each quadratic equation by completing the ...
 8.1.59: If g(x) = ax  2 5 b 2 , find all values of x for which g(x) = 9 25 .
 8.1.60: If g(x) = ax + 1 3 b 2 , find all values of x for which g(x) = 4 9 .
 8.1.61: If h(x) = 5(x + 2) 2 , find all values of x for which h(x) = 125.
 8.1.62: If h(x) = 3(x  4) 2 , find all values of x for which h(x) = 12.
 8.1.63: Three times the square of the difference between a number and 2 is ...
 8.1.64: Three times the square of the difference between a number and 9 is ...
 8.1.65: In Exercises 6568, solve the formula for the specified variable. Be...
 8.1.66: In Exercises 6568, solve the formula for the specified variable. Be...
 8.1.67: In Exercises 6568, solve the formula for the specified variable. Be...
 8.1.68: In Exercises 6568, solve the formula for the specified variable. Be...
 8.1.69: In Exercises 6972, solve each quadratic equation by completing the ...
 8.1.70: In Exercises 6972, solve each quadratic equation by completing the ...
 8.1.71: In Exercises 6972, solve each quadratic equation by completing the ...
 8.1.72: In Exercises 6972, solve each quadratic equation by completing the ...
 8.1.73: The ancient Greeks used a geometric method for completing the squar...
 8.1.74: An isosceles right triangle has legs that are the same length and a...
 8.1.75: In Exercises 7578, use the compound interest formula A = P(1 + r) t...
 8.1.76: In Exercises 7578, use the compound interest formula A = P(1 + r) t...
 8.1.77: In Exercises 7578, use the compound interest formula A = P(1 + r) t...
 8.1.78: In Exercises 7578, use the compound interest formula A = P(1 + r) t...
 8.1.79: a. According to the model, how many students, in thousands, were en...
 8.1.80: a. According to the model, how many students, in thousands, were en...
 8.1.81: The function s(t) = 16t 2 models the distance, s(t), in feet, that ...
 8.1.82: The function s(t) = 16t 2 models the distance, s(t), in feet, that ...
 8.1.83: Use the Pythagorean Theorem and the square root property to solve E...
 8.1.84: Use the Pythagorean Theorem and the square root property to solve E...
 8.1.85: Use the Pythagorean Theorem and the square root property to solve E...
 8.1.86: Use the Pythagorean Theorem and the square root property to solve E...
 8.1.87: Use the Pythagorean Theorem and the square root property to solve E...
 8.1.88: Use the Pythagorean Theorem and the square root property to solve E...
 8.1.89: A square flower bed is to be enlarged by adding 2 meters on each si...
 8.1.90: A square flower bed is to be enlarged by adding 4 feet on each side...
 8.1.91: What is the square root property?
 8.1.92: Explain how to solve (x  1) 2 = 16 using the square root property.
 8.1.93: Explain how to complete the square for a binomial. Use x2 + 6x to i...
 8.1.94: Explain how to solve x2 + 6x + 8 = 0 by completing the square.
 8.1.95: What is compound interest?
 8.1.96: In your own words, describe the compound interest formula A = P(1 +...
 8.1.97: Use a graphing utility to solve 4  (x + 1) 2 = 0. Graph y = 4  (x...
 8.1.98: Use a graphing utility to solve (x  1) 2  9 = 0. Graph y = (x  1...
 8.1.99: Use a graphing utility and x@intercepts to verify any of the real s...
 8.1.100: Make Sense? In Exercises 100103, determine whether each statement m...
 8.1.101: Make Sense? In Exercises 100103, determine whether each statement m...
 8.1.102: Make Sense? In Exercises 100103, determine whether each statement m...
 8.1.103: Make Sense? In Exercises 100103, determine whether each statement m...
 8.1.104: In Exercises 104107, determine whether each statement is true or fa...
 8.1.105: In Exercises 104107, determine whether each statement is true or fa...
 8.1.106: In Exercises 104107, determine whether each statement is true or fa...
 8.1.107: In Exercises 104107, determine whether each statement is true or fa...
 8.1.108: Solve for y: x2 a2 + y2 b2 = 1.
 8.1.109: Solve by completing the square: x2 + x + c = 0.
 8.1.110: Solve by completing the square: x2 + bx + c = 0.
 8.1.111: Solve: x4  8x2 + 15 = 0.
 8.1.112: Simplify: 4x  2  3[4  2(3  x)]. (Section 1.2, Example 14)
 8.1.113: Factor: 1  8x3 . (Section 5.5, Example 9)
 8.1.114: Divide: (x4  5x3 + 2x2  6) , (x  3). (Section 6.5, Example 1)
 8.1.115: Exercises 115117 will help you prepare for the material covered in ...
 8.1.116: Exercises 115117 will help you prepare for the material covered in ...
 8.1.117: Exercises 115117 will help you prepare for the material covered in ...
Solutions for Chapter 8.1: The Square Root Property and Completing the Square
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 8.1: The Square Root Property and Completing the Square
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1: The Square Root Property and Completing the Square includes 117 full stepbystep 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. Since 117 problems in chapter 8.1: The Square Root Property and Completing the Square have been answered, more than 29949 students have viewed full stepbystep solutions from this chapter.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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!).

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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