This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. Problem 25: Find the LCM of 6, 2xy^2, and 12x^2y. Step 1: Find the prime factorization of each term. 6 = 2 × 3 2xy^2 = 2 × x × y^2 12x^2y = 2^2 × 3 × x^2 × y Step 2: Identify the highest power of each prime factor and variable present in any of the terms. • For the prime factor 2: The highest power is 2^2 (from 12x^2y). • For the prime factor 3: The highest power is 3^1 (from 6 and 12x^2y). • For the variable x: The highest power is x^2 (from 12x^2y). • For the variable y: The highest power is y^2 (from 2xy^2). Step 3: Multiply these highest powers together to find the LCM. LCM = 2^2 × 3 × x^2 × y^2 = 4 × 3 × x^2 × y^2 = 12x^2y^2 The correct option is C. C. 12x^2y^2 Problem 26: Find x in the equation (2x)/(3) + (x)/(4) = 6. Step 1: Find a common denominator for the fractions. The least common multiple of 3 and 4 is 12. Multiply both sides of the equation by 12 to eliminate the denominators. 12 ( (2x)/(3) + (x)/(4) ) = 12(6) 12 ( (2x)/(3) ) + 12 ( (x)/(4) ) = 72 4(2x) + 3(x) = 72 Step 2: Simplify and solve for x. 8x + 3x = 72 11x = 72 x = (72)/(11) To express this as a mixed number: 72 ÷ 11 = 6 with a remainder of 6. x = 6(6)/(11) The correct option is C. C. 6(6)/(11) Problem 27: Factorize the expression xy - x + y - 1. Step 1: Group the terms. (xy - x) + (y - 1) Step 2: Factor out the common factor from each group. From the first group (xy - x), factor out x: x(y - 1). The second group is already (y - 1). x(y - 1) + 1(y - 1) Step 3: Factor out the common binomial factor (y - 1). (y - 1)(x + 1) The correct option is A. A. (x+1)(y-1) Problem 28: Factorize a^2 + 8a + 16. Step 1: Recognize the expression as a perfect square trinomial. A perfect square trinomial has the form x^2 + 2xy + y^2 = (x+y)^2. Here, a^2 is the first term, and 16 is 4^2. The middle term 8a is 2 × a × 4. Step 2: Write the expression in factored form. a^2 + 8a + 16 = (a + 4)^2 The correct option is C. C. (a+4)^2 Problem 29: Simplify 74^2 - 26^2. Step 1: Use the difference of squares formula, a^2 - b^2 = (a - b)(a + b). Here, a = 74 and b = 26. 74^2 - 26^2 = (74 - 26)(74 + 26) Step 2: Perform the subtraction and addition. 74 - 26 = 48 74 + 26 = 100 Step 3: Multiply the results. 48 × 100 = 4800 The correct option is A. A. 4800 Problem 30: Using the graph above, for what range of x is y < 0? Step 1: Identify the points where the graph intersects the x-axis (where y=0). From the graph, the parabola crosses the x-axis at x = -2 and x = 3. Step 2: Determine the region where the graph is below the x-axis. The parabola is below the x-axis between its x-intercepts. This means for values of x greater than -2 and less than 3. Step 3: Express this range as an inequality. -2 < x < 3 The correct option is A. A. -2 < x < 3 That's 2 down. 3 left today — send the next one.