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question:The quadratic (8x^2 48x 320) can be written in the form (a(x+b)^2 + c), where (a), (b), and (c) are constants. What is (a+b+c)?
answer:To write (8x^2 - 48x - 320) in the form (a(x+b)^2 + c), we first complete the square. 1. Factor out (8) from the first two terms: [ 8x^2 - 48x = 8(x^2 - 6x). ] 2. Determine the term to complete the square inside the parentheses, using ((x-3)^2 = x^2 - 6x + 9): [ 8(x^2 - 6x) = 8((x-3)^2 - 9). ] This simplifies to (8(x-3)^2 - 72). 3. Combine with the constant term (-320) from the original equation to match the desired form: [ 8(x-3)^2 - 72 - 320 = 8(x-3)^2 - 392. ] Therefore, the equation (8x^2 - 48x - 320) can be rewritten as: [ 8(x-3)^2 - 392. ] So, (a = 8), (b = -3), and (c = -392). Adding (a+b+c) results in: [ 8 - 3 - 392 = boxed{-387}. ] Conclusion: The given equation (8x^2 - 48x - 320) successfully transforms to the form (a(x+b)^2 + c) with (a = 8), (b = -3), and (c = -392), and the correct value of (a+b+c = -387).
question:Given a line l passing through the point P(3,4) and tangent to the circle x^{2}+y^{2}=25, find the equation of line l. A: y4= frac {4}{3}(x+3) B: y4= frac {3}{4}(x+3) C: y+4= frac {4}{3}(x3) D: y+4= frac {3}{4}(x3)
answer:It is evident that the point (-3,4) lies on the circle x^{2}+y^{2}=25. Let the slope of the tangent line be k, then the equation of the tangent line can be expressed as y-4=k(x+3), which simplifies to kx-y+3k-4=0. The distance d from the center of the circle (0,0) to the line is given by d= frac {|3k-4|}{sqrt {k^{2}+1}}=5 To find the value of k, we solve the equation which yields k= frac {3}{4}. Therefore, the equation of the tangent line is boxed{y-4= frac {3}{4}(x+3)} Hence, the correct answer is boxed{B} This problem examines the relationship between a line and a circle, involving knowledge of the point-slope form of a line's equation, the formula for the distance from a point to a line, and the general form of a line's equation. If a line is tangent to a circle, the distance from the circle's center to the line equals the circle's radius. Mastery of this property is key to solving this problem.
question:Given a>0, b>0, a+b=2 is a ( ) condition for ab≤1. A: Sufficient but not necessary condition B: Necessary but not sufficient condition C: Necessary and sufficient condition D: Neither sufficient nor necessary conditio
answer:Given a>0, b>0, and "a+b=2", we have (a+b)^2=4 which implies 4ab≤4 and thus "ab≤1" is correct. When a=10, b=0.1, we have ab≤1, but a+b=2 does not hold, which means the former implies the latter, but the latter does not imply the former, therefore, for a>0, b>0, "a+b=2" is a sufficient but not necessary condition for "ab≤1". Hence, the correct choice is boxed{A}.
question:The table below represents the percentage of students in each grade at three elementary schools: Annville, Cleona, and Brixton. [ begin{tabular}{rccccccc} & textbf{underline{K}} & textbf{underline{1}} & textbf{underline{2}} & textbf{underline{3}} & textbf{underline{4}} & textbf{underline{5}} & textbf{underline{6}} textbf{Annville:} & 18% & 14% & 13% & 15% & 14% & 16% & 10% textbf{Cleona:} & 10% & 14% & 16% & 16% & 14% & 20% & 10% textbf{Brixton:} & 20% & 15% & 15% & 12% & 12% & 11% & 15% end{tabular} ] Annville has 200 students, Cleona has 150 students, and Brixton has 180 students. In these three schools combined, what percent of the students are in Kindergarten? A) 12% B) 14% C) 16.42% D) 17% E) 18%
answer:1. **Calculate the number of Kindergarten students in each school:** - Annville: ( frac{18% times 200}{100} = 36 text{ students} ) - Cleona: ( frac{10% times 150}{100} = 15 text{ students} ) - Brixton: ( frac{20% times 180}{100} = 36 text{ students} ) 2. **Calculate the total number of Kindergarten students in all three schools:** - ( 36 + 15 + 36 = 87 text{ students} ) 3. **Calculate the combined total number of students in all three schools:** - ( 200 + 150 + 180 = 530 text{ students} ) 4. **Calculate the percentage of Kindergarten students in the combined student population:** - ( frac{87}{530} times 100% approx 16.42% ) 5. **Conclusion:** - The percent of Kindergarten students in the three schools combined is ( 16.42% ). The final answer is boxed{C) 16.42%}