Simplify the expression sqrt(100) - 72 / 24. Solve the equation (x+2)/(x+3) = (1)/(x-2). Given y = sqrt(x2 + 1), find (dy)/(dx).

Here are the solutions to the problems you sent. Problem 1: Simplify the expression sqrt(100) - 72 ÷ 24. Step 1: Calculate the square root. sqrt(100) = 10 Step 2: Perform the division. 72 ÷ 24 = 3 Step 3: Perform the subtraction. 10 - 3 = 7 The simplified expression is 7. --- Problem 2: Solve the equation (x+2)/(x+3) = (1)/(x-2). Step 1: Cross-multiply the terms. (x+2)(x-2) = 1(x+3) Step 2: Expand both sides of the equation. x^2 - 4 = x + 3 Step 3: Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. x^2 - x - 4 - 3 = 0 x^2 - x - 7 = 0 Step 4: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a to solve for x. Here, a=1, b=-1, c=-7. x = -(-1) ± sqrt((-1)^2 - 4(1)(-7))2(1) x = 1 ± sqrt(1 + 28)2 x = 1 ± sqrt(29)2 The solutions for x are 1 ± sqrt(29)2. --- Problem 3: Given y = sqrt(x^2 + 1), find (dy)/(dx). Step 1: Rewrite the function using exponent notation. y = (x^2 + 1)^1/2 Step 2: Apply the chain rule. Let u = x^2 + 1. Then y = u^1/2. The derivative of y with respect to u is: (dy)/(du) = (1)/(2) u^-1/2 The derivative of u with respect to x is: (du)/(dx) = 2x Step 3: Multiply the derivatives (dy)/(du) and (du)/(dx) to find (dy)/(dx). (dy)/(dx) = (1)/(2) (x^2 + 1)^-1/2 · (2x) (dy)/(dx) = x (x^2 + 1)^-1/2 Step 4: Rewrite the expression with a positive exponent and radical form. (dy)/(dx) = (x)/(sqrt(x^2 + 1)) The derivative is (x)/(sqrt(x^2 + 1)). --- Problem 4: Simplify the expression y = (x^2 - 3x + 2)/(x^2 - 1). Step 1: Factor the numerator x^2 - 3x + 2. We look for two numbers that multiply to 2 and add to -3. These numbers are -1 and -2. x^2 - 3x + 2 = (x-1)(x-2) Step 2: Factor the denominator x^2 - 1. This is a difference of squares, a^2 - b^2 = (a-b)(a+b). x^2 - 1 = (x-1)(x+1) Step 3: Substitute the factored forms back into the expression. y = ((x-1)(x-2))/((x-1)(x+1)) Step 4: Cancel out the common factor (x-1), assuming x ≠ 1. y = (x-2)/(x+1) The simplified expression is (x-2)/(x+1). Send me the next one 📸