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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5.1 2x^2 = -x Step 1: Move all terms to one side to set the equation to zero. 2x^2 + x = 0 Step 2: Factor out the common term x. x(2x + 1) = 0 Step 3: Apply the Zero Factor Property. x = 0 or 2x + 1 = 0 2x = -1 x = -(1)/(2) The solutions are x = 0 or x = -(1)/(2). 5.2 x^2 - 25 = 0 Step 1: Add 25 to both sides of the equation. x^2 = 25 Step 2: Take the square root of both sides. Remember to include both positive and negative roots. sqrt(x^2) = ±sqrt(25) x = ± 5 The solutions are x = 5 or x = -5. 5.3 x^2 + 5x - 6 = 0 Step 1: Factor the quadratic expression. Find two numbers that multiply to -6 and add to 5. These numbers are 6 and -1. (x + 6)(x - 1) = 0 Step 2: Apply the Zero Factor Property. x + 6 = 0 or x - 1 = 0 x = -6 or x = 1 The solutions are x = -6 or x = 1. 5.4 (x+5)/(3) + (x+1)/(6) = (x)/(3) - 2 Step 1: Find the least common denominator (LCD) for all fractions, which is 6. Multiply every term by the LCD to clear the denominators. 6 ((x+5)/(3)) + 6 ((x+1)/(6)) = 6 ((x)/(3)) - 6(2) 2(x+5) + (x+1) = 2x - 12 Step 2: Distribute and simplify both sides of the equation. 2x + 10 + x + 1 = 2x - 12 3x + 11 = 2x - 12 Step 3: Gather x terms on one side and constant terms on the other. 3x - 2x = -12 - 11 x = -23 The solution is x = -23. 5.5 An amount of R34,80 is made up of 50c and 20c coins. How many 50c coins are there out of a total of 120 coins? Step 1: Define variables and set up the equations. Let f be the number of 50c coins. Let t be the number of 20c coins. Total number of coins: f + t = 120 (1) Total value (in Rands): 0.50f + 0.20t = 34.80 (2) Step 2: Solve equation (1) for t. t = 120 - f (3) Step 3: Substitute equation (3) into equation (2). 0.50f + 0.20(120 - f) = 34.80 Step 4: Distribute and solve for f. 0.50f + 24 - 0.20f = 34.80 0.30f + 24 = 34.80 0.30f = 34.80 - 24 0.30f = 10.80 f = (10.80)/(0.30) f = 36 There are 36 50c coins. Last free one today — make it count tomorrow, or type /upgrade for unlimited.