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
Ãrvela, let's knock this out. Quadratic Equations 1. Solve: 3x^2 - 7x + 2 = 0 Step 1: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here a=3, b=-7, c=2. x = -(-7) ± sqrt((-7)^2 - 4(3)(2))2(3) x = 7 ± sqrt(49 - 24)6 x = 7 ± sqrt(25)6 x = (7 ± 5)/(6) Step 2: Calculate the two solutions. x_1 = (7 + 5)/(6) = (12)/(6) = 2 x_2 = (7 - 5)/(6) = (2)/(6) = (1)/(3) The solutions are x = 2, x = (1)/(3). 2. Solve using completing the square: x^2 + 6x + 5 = 0 Step 1: Move the constant term to the right side. x^2 + 6x = -5 Step 2: Complete the square by adding ((6)/(2))^2 = 3^2 = 9 to both sides. x^2 + 6x + 9 = -5 + 9 (x + 3)^2 = 4 Step 3: Take the square root of both sides. x + 3 = ±sqrt(4) x + 3 = ± 2 Step 4: Solve for x. x = -3 ± 2 x_1 = -3 + 2 = -1 x_2 = -3 - 2 = -5 The solutions are x = -1, x = -5. Surds and Indices 1. Simplify: sqrt(50) + 3sqrt(8) Step 1: Simplify each surd. sqrt(50) = sqrt(25 × 2) = 5sqrt(2) sqrt(8) = sqrt(4 × 2) = 2sqrt(2) Step 2: Substitute and combine like terms. 5sqrt(2) + 3(2sqrt(2)) 5sqrt(2) + 6sqrt(2) 11sqrt(2) The simplified expression is 11sqrt(2). 2. Simplify: (2^3 × 2^-5) Step 1: Use the index rule a^m × a^n = a^m+n. 2^3 + (-5) 2^3 - 5 2^-2 Step 2: Use the index rule a^-n = (1)/(a^n). (1)/(2^2) (1)/(4) The simplified expression is (1)/(4). Logarithms 1. Simplify: _10(100) + _10(0.001) Step 1: Evaluate each logarithm. _10(100) = 2 (since 10^2 = 100) _10(0.001) = -3 (since 10^-3 = 0.001) Step 2: Add the values. 2 + (-3) = -1 The simplified expression is -1. 2. Solve: _2(x) = 5 Step 1: Convert the logarithmic equation to an exponential equation using the definition _b(x) = y b^y = x. x = 2^5 Step 2: Calculate the value. x = 32 The solution is x = 32. Trigonometry 1. Solve: x = 1/2 for 0^ x 360^ Step 1: Find the reference angle. ^-1((1)/(2)) = 30^ Step 2: Identify the quadrants where x is positive (Quadrant I and Quadrant II).