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 to the problems: 1. Solve the equation 2x^2 - 13x + 20 = 0. Step 1: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=2, b=-13, c=20. x = -(-13) ± sqrt((-13)^2 - 4(2)(20))2(2) Step 2: Simplify the expression under the square root. x = 13 ± sqrt(169 - 160)4 x = 13 ± sqrt(9)4 Step 3: Calculate the square root. x = (13 ± 3)/(4) Step 4: Find the two possible values for x. x_1 = (13 + 3)/(4) = (16)/(4) = 4 x_2 = (13 - 3)/(4) = (10)/(4) = (5)/(2) The solutions are: x = 4 or x = (5)/(2) 2. A delivery company models fuel consumption using the rule: F = (64a^7b^9)/(8a^3b^3). Simplify using index laws. Step 1: Separate the numerical coefficients and the variables. F = ((64)/(8)) × ((a^7)/(a^3)) × ((b^9)/(b^3)) Step 2: Simplify the numerical coefficient and apply the index law (x^m)/(x^n) = x^m-n for the variables. F = 8 × a^7-3 × b^9-3 Step 3: Perform the subtractions in the exponents. F = 8a^4b^6 3. If a=2 and b=3, find the numerical value of the simplified expression from Problem 2. Step 1: Substitute a=2 and b=3 into the simplified expression F = 8a^4b^6. F = 8(2)^4(3)^6 Step 2: Calculate the powers. F = 8(16)(729) Step 3: Multiply the values. F = 128 × 729 F = 93312 4. A student plans to fence a rectangular kitchen garden whose area is given by A = x^2 - 4x - 21. Factorize. Step 1: Find two numbers that multiply to -21 and add up to -4. These numbers are 3 and -7. Step 2: Rewrite the quadratic expression using these numbers. A = x^2 + 3x - 7x - 21 Step 3: Factor by grouping. A = x(x + 3) - 7(x + 3) Step 4: Factor out the common binomial factor. A = (x + 3)(x - 7) Got more? Send 'em!