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. *a) Solve 3x(x+5)=0 Step 1: Set each factor equal to zero. 3x = 0 or x+5 = 0 Step 2: Solve for x in each equation. x = (0)/(3) or x = -5 x = 0 or x = -5 The solutions are x=0, x=-5. *b) Solve 5x(2x-7)=0 Step 1: Set each factor equal to zero. 5x = 0 or 2x-7 = 0 Step 2: Solve for x in each equation. x = (0)/(5) or 2x = 7 x = 0 or x = (7)/(2) The solutions are x=0, x=(7)/(2). *c) Solve 2x^2 - 6x = 0 Step 1: Factor out the common term, 2x. 2x(x-3) = 0 Step 2: Set each factor equal to zero. 2x = 0 or x-3 = 0 Step 3: Solve for x in each equation. x = (0)/(2) or x = 3 x = 0 or x = 3 The solutions are x=0, x=3. *d) Solve 3x^2 - 15x = 0 Step 1: Factor out the common term, 3x. 3x(x-5) = 0 Step 2: Set each factor equal to zero. 3x = 0 or x-5 = 0 Step 3: Solve for x in each equation. x = (0)/(3) or x = 5 x = 0 or x = 5 The solutions are x=0, x=5. *e) Solve 6x^2 + 10x = 0 Step 1: Factor out the common term, 2x. 2x(3x+5) = 0 Step 2: Set each factor equal to zero. 2x = 0 or 3x+5 = 0 Step 3: Solve for x in each equation. x = (0)/(2) or 3x = -5 x = 0 or x = -(5)/(3) The solutions are x=0, x=-(5)/(3). *f) Solve 4x^2 + 20x = 0 Step 1: Factor out the common term, 4x. 4x(x+5) = 0 Step 2: Set each factor equal to zero. 4x = 0 or x+5 = 0 Step 3: Solve for x in each equation. x = (0)/(4) or x = -5 x = 0 or x = -5 The solutions are x=0, x=-5. *g) Solve 6x^2 + 2x = 0 Step 1: Factor out the common term, 2x. 2x(3x+1) = 0 Step 2: Set each factor equal to zero. 2x = 0 or 3x+1 = 0 Step 3: Solve for x in each equation. x = (0)/(2) or 3x = -1 x = 0 or x = -(1)/(3) The solutions are x=0, x=-(1)/(3). *h) Solve 10x - 18x^2 = 0 Step 1: Factor out the common term, 2x. 2x(5-9x) = 0 Step 2: Set each factor equal to zero. 2x = 0 or 5-9x = 0 Step 3: Solve for x in each equation. x = (0)/(2) or 5 = 9x x = 0 or x = (5)/(9) The solutions are x=0, x=(5)/(9). *i) Solve 8x^2 = 6x Step 1: Move all terms to one side to set the equation to zero. 8x^2 - 6x = 0 Step 2: Factor out the common term, 2x. 2x(4x-3) = 0 Step 3: Set each factor equal to zero. 2x = 0 or 4x-3 = 0 Step 4: Solve for x in each equation. x = (0)/(2) or 4x = 3 x = 0 or x = (3)/(4) The solutions are x=0, x=(3)/(4). *j) Solve 10x^2 = 15x Step 1: Move all terms to one side to set the equation to zero. 10x^2 - 15x = 0 Step 2: Factor out the common term, 5x. 5x(2x-3) = 0 Step 3: Set each factor equal to zero. 5x = 0 or 2x-3 = 0 Step 4: Solve for x in each equation. x = (0)/(5) or 2x = 3 x = 0 or x = (3)/(2) The solutions are x=0, x=(3)/(2).