6. In Chapter V of Ars Magna, Cardano gives formulas for solving the different forms of quadratic equations. (This is basically what al-Khwarizmi did in his algebra book except Cardano's formulas yield both solutions of quadratic equations.) Cardano's Rule III is for quadratic equations of the form where the first power is equal to the square and number. Let N be the number, a is the coefficient of the first power, and the coefficient of the square is 1. Here is Cardano's rhetorical solution: Multiply one-half the coefficient of the first power by itself and, having subtracted the number from the product, subtract the root of the remainder from one-half the coefficient of the first power or add the two of them, and the value of x will be both the sum and the difference.

5) On Monday an electric bass craftsperson attaches frets to four necks and cuts three frames. Doing all of this takes a total of 6 hours, 10 minutes. On Tuesday this same craftsperson spends a total of seven hours attaching frets to three necks and cutting seven frames. On Wednesday this craftsperson has to attach frets to one neck and cut four frames. How long should this take? Answer in hours and minutes. Write two (or three) algebra equations, then solve. Assume that each type of task always takes the same amount of time.

1. Solve the following: a) x² – 9x = 0 b) x² + 4x – 21 = 0 c) x2 – 25 = 0

Express (3 + 4√ 2)-1 in the form a + b√ 2, where a, b ∈Q.