answer:The real solutions to the given equation are: [ x = frac{1}{102} left(-135 - 7 sqrt{365}right), quad x = frac{1}{102} left(-135 + 7 sqrt{365}right) ]

answer:The exact value of #sin^(-1)(1/2)# can be expressed as #theta=30^@ + 360n^@# and #theta=150^@ + 360n^@#, where #n# is an integer. Explanation: The inverse sine function, #sin^(-1)#, aims to find an angle whose sine is a given value. In this case, we're looking for an angle #theta# where #sin(theta) = 1/2#. We know that in a 30-60-90 right triangle, the sine of the 30-degree angle is 1/2 because the ratio of the opposite side to the hypotenuse is 1:2. Since sine is positive in the first and second quadrants, we have two possible angles: one in the first quadrant, which is #30^@#, and one in the second quadrant, which is #180^@ - 30^@ = 150^@#. Adding multiples of 360 degrees accounts for all possible angles in the same sine value, hence the #360n^@# term. Therefore, the solutions are #theta=30^@ + 360n^@# and #theta=150^@ + 360n^@#, where #n# is an integer.

answer:The purchase of a building acquires an asset and this does not affect the liabilities or equity Dr. Carter has as the asset of cash is reduced to credit the action. The following entry is made: DateAccountDebitCredit mm/ddBuilding500,000 Cash 500,000 Note:To record the purchase of a building

answer:The roots of the polynomial f(x) = x^2 - 9x + 6 in mathbb{Z}_{14} can be found by using the quadratic formula, since mathbb{Z}_{14} is not a field. We have: x equiv frac{9 pm sqrt{9^2 - 4 cdot 1 cdot 6}}{2} equiv frac{9 pm sqrt{1}}{2} pmod{14} Since 1 has two square roots in mathbb{Z}_{14}, 1 and 13, we get two pairs of roots: 1. x equiv frac{9 + 1}{2} equiv 5 pmod{14} and x equiv frac{9 - 13}{2} equiv 12 pmod{14} 2. x equiv frac{9 + 13}{2} equiv 11 pmod{14} and x equiv frac{9 - 1}{2} equiv 4 pmod{14} This gives us the factorizations: f(x) = (x - 5)(x - 12) = (x - 11)(x - 4) pmod{14} For the polynomial g(x) = x^2 - 9x + 3 in mathbb{Z}_{17}, since mathbb{Z}_{17} is a field, the quadratic formula applies: x equiv frac{9 pm sqrt{9^2 - 4 cdot 1 cdot 3}}{2} equiv frac{9 pm sqrt{1}}{2} pmod{17} The square root of 1 in mathbb{Z}_{17} is 1 itself, so we have: x equiv frac{9 pm 1}{2} equiv 5 text{ and } 4 pmod{17} Hence, the factorization is: g(x) = (x - 5)(x - 4) pmod{17} In summary: 1. f(x) in mathbb{Z}_{14} factors as (x - 5)(x - 12) = (x - 11)(x - 4). 2. g(x) in mathbb{Z}_{17} factors as (x - 5)(x - 4). These factorizations represent all possible monic factor pairs for the given polynomials in the respective rings.