answer:Definitions and Properties - The logarithm function, denoted as (log_b n), is the value of (x) that satisfies (n = b^x). Here (b) is the base of the logarithm. - The logarithm of a power can be expressed as: (y log_b n = log_b n^y). Solving the Equation Starting with the equation: [ log_2 (x+3) + 2 log_2 5 = 4 ] 1. **Apply the power rule of logarithms**: [ log_2 (x+3) + log_2 5^2 = 4 ] Using (y log_b n = log_b n^y), we rewrite (2 log_2 5) as (log_2 25). 2. **Combine the logarithms**: [ log_2 [(x+3) times 25] = 4 ] Using the property (log_b x + log_b y = log_b (xy)). 3. **Solve for (x)**: [ (x+3) times 25 = 2^4 ] Exponentiating both sides with base 2, knowing that (2^4 = 16). 4. **Isolate (x)**: [ x+3 = frac{16}{25} ] [ x = frac{16}{25} - 3 = frac{16}{25} - frac{75}{25} = frac{-59}{25} ] Thus, the solution to the equation (log_2 (x+3) + 2 log_2 5 = 4) is: [ frac{-59{25}} ] boxed{The final answer is (boxed{A})}

answer:In triangle ABC, since AD is a median, D is the midpoint of BC, thus BD = CD = 5. Let P be the projection of point A onto line BC, and let x = BP, hence PD = 5 - x. Let h = AP. Appling the Pythagorean theorem on right triangles APB, APC, and APD, we derive the equations: begin{align*} AB^2 &= x^2 + h^2, AC^2 &= (10 - x)^2 + h^2, 36 &= (5 - x)^2 + h^2. end{align*} Adding AB^2 and AC^2, we find: [ AB^2 + AC^2 = x^2 + h^2 + (10 - x)^2 + h^2 = 2x^2 - 20x + 100 + 2h^2. ] By substituting h^2 from 36 = (5 - x)^2 + h^2, h^2 = 36 - (5-x)^2, we get: [ h^2 = 36 - (25 - 10x + x^2), ] [ h^2 = 11 + 10x - x^2, ] and thus [ AB^2 + AC^2 = 2x^2 - 20x + 100 + 2(11 + 10x - x^2) = 122. ] Thus, the value of AB^2 + AC^2 is boxed{122}.

answer:To find when the parabolas intersect, we set 3x^2 - 6x + 6 = -2x^2 + x + 3. This leads to: [3x^2 - 6x + 6 = -2x^2 + x + 3,] [5x^2 - 7x + 3 = 0.] Apply the quadratic formula, x = frac{-b pm sqrt{b^2 - 4ac}}{2a}, where a = 5, b = -7, and c = 3: [x = frac{-(-7) pm sqrt{(-7)^2 - 4 cdot 5 cdot 3}}{2 cdot 5} = frac{7 pm sqrt{49 - 60}}{10} = frac{7 pm sqrt{-11}}{10}.] Since the square root of a negative number does not produce a real number, the square root term sqrt{-11} indicates there are no real intersections between the parabolas. Conclusion: Since we found no real solutions for x, the parabolas do not intersect in real coordinates. Thus, it is not possible to find c and a or compute c-a. Therefore, boxed{text{no real solutions for } c - a}.

answer:If there are 1 million moose in Canada and for every moose there are two beavers, then there are 2 million beavers in Canada. Now, we have 38 million people and 2 million beavers. To find the ratio of humans to beavers, we divide the number of people by the number of beavers: 38 million people / 2 million beavers = 19 people per beaver So the ratio of humans to beavers is boxed{19:1} .