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.
ScanSolve AI Answer
Here are the solutions for the 1st question: 1. Multiply the binomials. 1 (i): (2x+5) and (4x-3) Step 1: Use the distributive property (FOIL method). (2x+5)(4x-3) = 2x(4x) + 2x(-3) + 5(4x) + 5(-3) Step 2: Perform the multiplications. = 8x^2 - 6x + 20x - 15 Step 3: Combine like terms. = 8x^2 + 14x - 15 1 (ii): (y-8) and (3y-4) Step 1: Use the distributive property (FOIL method). (y-8)(3y-4) = y(3y) + y(-4) - 8(3y) - 8(-4) Step 2: Perform the multiplications. = 3y^2 - 4y - 24y + 32 Step 3: Combine like terms. = 3y^2 - 28y + 32 1 (iii): (2.5l - 0.5m) and (2.5l + 0.5m) Step 1: Recognize this as a difference of squares: (A-B)(A+B) = A^2 - B^2. Here A=2.5l and B=0.5m. (2.5l - 0.5m)(2.5l + 0.5m) = (2.5l)^2 - (0.5m)^2 Step 2: Square each term. = (2.5)^2 l^2 - (0.5)^2 m^2 = 6.25l^2 - 0.25m^2 1 (iv): (a+3b) and (x+5) Step 1: Use the distributive property (FOIL method). (a+3b)(x+5) = a(x) + a(5) + 3b(x) + 3b(5) Step 2: Perform the multiplications. = ax + 5a + 3bx + 15b 1 (v): (2pq+3q^2) and (3pq-2q^2) Step 1: Use the distributive property (FOIL method). (2pq+3q^2)(3pq-2q^2) = 2pq(3pq) + 2pq(-2q^2) + 3q^2(3pq) + 3q^2(-2q^2) Step 2: Perform the multiplications. = 6p^2q^2 - 4pq^3 + 9pq^3 - 6q^4 Step 3: Combine like terms. = 6p^2q^2 + 5pq^3 - 6q^4 1 (vi): ((3)/(4)a^2+3b^2) and 4(a^2-(2)/(3)b^2) Step 1: Distribute the 4 into the second expression. 4(a^2-(2)/(3)b^2) = 4a^2 - 4 × (2)/(3)b^2 = 4a^2 - (8)/(3)b^2 Step 2: Multiply the two binomials: ((3)/(4)a^2+3b^2)(4a^2-(8)/(3)b^2). = (3)/(4)a^2(4a^2) + (3)/(4)a^2(-(8)/(3)b^2) + 3b^2(4a^2) + 3b^2(-(8)/(3)b^2) Step 3: Perform the multiplications. = 3a^4 - (24)/(12)a^2b^2 + 12a^2b^2 - (24)/(3)b^4 Step 4: Simplify the fractions and combine like terms. = 3a^4 - 2a^2b^2 + 12a^2b^2 - 8b^4 = 3a^4 + 10a^2b^2 - 8b^4 3 done, 2 left today. You're making progress.