Math 021 Formulas and Concepts (Part 3 of 3)Rate of work: The number of jobs that can be completed in a given unit of time. If one job can be completed in units of time, then the rate of work is given by 1/tRate for two workers: 1/t1 + 1/t2 = 1/tHere, and are the individual times to complete one job, and is the time to complete the job when working together. Definition Principal and Negative Square RootsA non-negative real number b is the principal square root of a non-negative real number a,denoted as b = √a , if b2 = a .A negative real number b is the negative square root of a non-negative real number a,denoted as b =  √a , if b2 = a .√a2 = |a|Simplifying Radical Expressions of the Form n√anIf n is an integer such that n ≥2 and a is a real number, then n√an  a if n is odd.n√an  |a| if n is evenDomain of a Radical FunctionIf the index of a radical function is even, the radicand must be greater than or equal to zero.If the index of a radical function is odd, the radicand can be any real number.Rationalizing a Denominator with One TermTo rationalize a denominator with a single radical of index n, multiply the numerator and denominator by a radical of index n so that the radicand in the denominator is a perfect nth power.Rationalizing a Denominator with Two TermsTo rationalize a denominator with two terms involving one or more square roots, multiply the numerator and denominator by the conjugate of the denominator.Imaginary Unit ii = √-1i2 = -1Simplifying i: divide n by 4Remainder0 1 2 3i n1 i -1 -iComplex NumbersThe set of all numbers of the form: a bi √-a = √-1 √a = i√aSquare Root PropertyU is an algebraic Expression, K € Real number (can be +/-)U2 = K  U = √K or U = -√KEvery quadratic equation can be written in the form: (x-h)2 = K(x-h)2 = K  x-h = +/- √KQuadratic Formulax = -b +/- √b 2 - 4ac 2aDiscriminantD > 0 = 2 real solutionsD < 0 = 2 non-real solutionsD=0  1 real solutionQuadratic Function Second-degree polynomial function of the form f(x) = ax2 + bx +c Where a, b, and c are real numbers and a ≠ 0Every quadratic function has a “u-shaped” graph called a parabola.If a  0, the parabola opens down and the function has a maximum value at the vertex.If a  0, the parabola opens down and the function has a minimum value at the vertex.Distance formula (for a plane)The distance between any two points A(x1,y1) and B(x2,y1) is given by the formula:d(A,B) = √(x2 - x1)2 (y2 - y1)2Midpoint FormulaThe midpoint of the line segment from A(x1,y1) and B(x2,y1) is:(x2 + x1) , (y2 + y1) 2 2CircleThe set of all points (x, y) in the Cartesian plane that are a fixed distance r from a fixedpoint (h,k) The fixed distance r is the radius of the circle The fixed point (h,k) is the center of the circleStandard form Circle EquationStandard form of an equation of a circle with center (h,k) and radius r:(x-h)2 + (y-k)2 = r2The standard form of an equation of a circle centered at the origin with radius r:x2 + y2 =