answer:1. **Identify the form of the quadratic equation**: The given quadratic equation is x^2 + 2xsqrt{3} + 3 = 0. 2. **Recall the general form of a quadratic equation**: A quadratic equation is generally written as ax^2 + bx + c = 0. Comparing this with the given equation, we have a = 1, b = 2sqrt{3}, and c = 3. 3. **Calculate the discriminant**: The discriminant Delta of a quadratic equation ax^2 + bx + c = 0 is given by Delta = b^2 - 4ac. Plugging in the values from our equation: [ Delta = (2sqrt{3})^2 - 4 cdot 1 cdot 3 = 12 - 12 = 0. ] 4. **Interpret the discriminant**: A discriminant of zero implies that the quadratic equation has exactly one distinct real root, or equivalently, two real and equal roots. 5. **Find the roots using the quadratic formula**: The roots of the quadratic equation can be found using the formula x = frac{-b pm sqrt{Delta}}{2a}. Since Delta = 0, this simplifies to: [ x = frac{-2sqrt{3}}{2 cdot 1} = -sqrt{3}. ] Thus, the equation has a double root at x = -sqrt{3}. 6. **Conclusion**: Since the roots are real and equal, the correct answer is: [ boxed{textbf{(A)} text{real and equal}} ]

answer:Solution: f(x)=sin^{2}(x+ frac {pi}{4})+cos^{2}(x+ frac {pi}{4}- frac {pi}{2})-1=2sin^{2}(x+ frac {pi}{4})-1=-cos(2x+ frac {pi}{2}) =sin(2x), Therefore, T= frac {2pi}{2}=pi, Hence, the correct option is boxed{A}. First, simplify using the double angle formula and the co-function identities, and then determine the answer by combining the smallest positive period T= frac {2pi}{omega} and the odd-even properties of the sine function. This question mainly examines the application of the double angle formula and co-function identities, as well as the basic properties of trigonometric functions—the smallest positive period and odd-even properties. Trigonometric formulas are numerous and not easy to remember; only with regular accumulation and practice can one apply them proficiently in exams.

answer:Let's calculate the cost of the red roses first. Since Jezebel needs two dozens of red roses and there are 24 roses in two dozens (12 roses in a dozen), we can calculate the cost as follows: Cost of red roses = Number of red roses * Cost per red rose Cost of red roses = 24 * 1.50 Cost of red roses = 36 Now, let's find out how much money Jezebel has left for the sunflowers after buying the red roses: Money left for sunflowers = Total money - Cost of red roses Money left for sunflowers = 45 - 36 Money left for sunflowers = 9 Finally, we can calculate the number of sunflowers Jezebel can buy with the remaining money: Number of sunflowers = Money left for sunflowers / Cost per sunflower Number of sunflowers = 9 / 3 Number of sunflowers = 3 Jezebel needs to buy boxed{3} sunflowers.

answer:1. **Identify the speeds and distances**: - Let ( v_S ), ( v_L ), and ( v_K ) be the speeds of Sasha, Lesha, and Kolya, respectively. - Given that Kolya's speed is 0.9 times Lesha's speed: ( v_K = 0.9 v_L ). 2. **Determine Lesha's distance when Sasha finishes**: - When Sasha finishes the 100 meters run, Lesha is 10 meters behind. - Thus, Lesha has run 90 meters by the time Sasha finishes: [ d_L = 90 text{ meters} ] 3. **Calculate the time it took for Sasha to finish**: - Let ( t ) be the time for Sasha to finish the race. - Using the distance formula: ( d_S = v_S cdot t ) where ( d_S = 100 ) meters: [ 100 = v_S cdot t Rightarrow t = frac{100}{v_S} ] 4. **Relate the time taken by Lesha and Kolya to their distances**: - Lesha's distance in this time ( t ): [ d_L = v_L cdot t = 90 text{ meters} ] - Substitute ( t ) from Sasha's time: [ v_L cdot frac{100}{v_S} = 90 Rightarrow v_L = frac{90 v_S}{100} = 0.9 v_S ] 5. **Find Kolya's distance when Sasha finishes**: - Kolya's distance in the same time ( t ): [ d_K = v_K cdot t = 0.9 v_L cdot t = 0.9 (0.9 v_S) cdot frac{100}{v_S} = 0.81 cdot 100 = 81 text{ meters} ] 6. **Determine the distance between Sasha and Kolya**: - The distance between where Sasha and Kolya are when Sasha finishes is: [ 100 text{ meters} - 81 text{ meters} = 19 text{ meters} ] # Conclusion The distance between Sasha and Kolya when Sasha finishes is: [ boxed{19 text{ meters}} ]