answer:For a quadratic equation in the form ax^2 + bx + c = 0, the product of the roots can be found using the formula frac{c}{a}. Given the equation 24x^2 + 36x - 648 = 0, we identify: - a = 24 - c = -648 Using the formula: [ text{Product of the roots} = frac{c}{a} = frac{-648}{24} = -27 ] Thus, the product of the roots of the equation 24x^2 + 36x - 648 = 0 is boxed{-27}.

answer:To find the set of all values of x that satisfy the inequality 2x - 3 > 7 - x, we start by simplifying the inequality: 1. **Combine like terms**: [ 2x - 3 > 7 - x ] Add x to both sides to get all x terms on one side: [ 2x + x > 7 + 3 ] Simplify: [ 3x > 10 ] 2. **Solve for x**: Divide both sides by 3 to isolate x: [ x > frac{10}{3} ] This tells us that x must be greater than frac{10}{3}. Therefore, the correct answer is: [ boxed{textbf{(D)} x > frac{10}{3}} ]

answer:: 1. Consider the two triplets (17, 18, 19) and (20, 21, 22). 2. We need to compare the least common multiple (LCM or НАОК in Russian) of each set of numbers. 3. **Calculate text{LCM}(17, 18, 19)**: * Since 17 and 19 are prime numbers, their greatest common divisor (GCD) with any other number is 1. * The LCM of any set of numbers is the product of their highest power factors: [ text{LCM}(17, 18, 19) = 17 cdot 18 cdot 19 ] Detailed calculation: [ 17 times 18 = 306 ] [ 306 times 19 = 5814 ] Hence: [ text{LCM}(17, 18, 19) = 5814 ] 4. **Calculate text{LCM}(20, 21, 22)**: * Factorize each number: * 20 = 2^2 cdot 5 * 21 = 3 cdot 7 * 22 = 2 cdot 11 * The LCM is the product of the highest powers of each prime that appears: [ text{LCM}(20, 21, 22) = 2^2 cdot 3 cdot 5 cdot 7 cdot 11 ] Detailed calculation: [ 2^2 = 4 ] [ 4 times 3 = 12 ] [ 12 times 5 = 60 ] [ 60 times 7 = 420 ] [ 420 times 11 = 4620 ] Hence: [ text{LCM}(20, 21, 22) = 4620 ] 5. **Conclusion**: We have: [ text{LCM}(17, 18, 19) = 5814 ] [ text{LCM}(20, 21, 22) = 4620 ] Therefore, text{LCM}(17, 18, 19) > text{LCM}(20, 21, 22). [ boxed{text{Существуют}} ]

answer:In the equation ( sqrt {3}x-y+3=0), let (y=0) to get (x=- sqrt {3}). (∴M(- sqrt {3},0)), The slope of the line ( sqrt {3}x-y+3=0) is ( sqrt {3}), so its inclination angle is (60^{circ}). When line (l) is rotated around point (M) by (30^{circ}), if it is counterclockwise, then the inclination angle of line (l') is (90^{circ}), (∴) The equation of line (l') is (x=- sqrt {3}); If it is clockwise, then the inclination angle of line (l') is (30^{circ}), (∴) The slope of line (l') is ( frac { sqrt {3}}{3}), (∴) The equation of line (l') is (y-0= frac { sqrt {3}}{3}(x+ sqrt {3})), which simplifies to (x- sqrt {3}y+ sqrt {3}=0). Thus, the equations of the resulting line (l') are (x=- sqrt {3}) and (x- sqrt {3}y+ sqrt {3}=0), depending on the direction of rotation. ( boxed{text{The equations of line } l' text{ are } x=- sqrt {3} text{ and } x- sqrt {3}y+ sqrt {3}=0.} )