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Simplifying 2(3x + 5) = 3x(4x + 5) Reorder the terms: 2(5 + 3x) = 3x(4x + 5) (5 * 2 + 3x * 2) = 3x(4x + 5) (10 + 6x) = 3x(4x + 5) Reorder the terms: 10 + 6x = 3x(5 + 4x) 10 + 6x = (5 * 3x + 4x * 3x) 10 + 6x = (15x + 12x2) Solving 10 + 6x = 15x + 12x2 Solving for variable 'x'. Combine like terms: 6x + -15x = -9x 10 + -9x + -12x2 = 15x + 12x2 + -15x + -12x2 Reorder the terms: 10 + -9x + -12x2 = 15x + -15x + 12x2 + -12x2 Combine like terms: 15x + -15x = 0 10 + -9x + -12x2 = 0 + 12x2 + -12x2 10 + -9x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 10 + -9x + -12x2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -0.8333333333 + 0.75x + x2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 0.75x + 0.8333333333 + x2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 0.75x + x2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 0.75x + x2 = 0 + 0.8333333333 0.75x + x2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 0.75x + x2 = 0.8333333333 The x term is 0.75x. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75x + 0.140625 + x2 = 0.8333333333 + 0.140625 Reorder the terms: 0.140625 + 0.75x + x2 = 0.8333333333 + 0.140625 Combine like terms: 0.8333333333 + 0.140625 = 0.9739583333 0.140625 + 0.75x + x2 = 0.9739583333 Factor a perfect square on the left side: (x + 0.375)(x + 0.375) = 0.9739583333 Calculate the square root of the right side: 0.986893274 Break this problem into two subproblems by setting (x + 0.375) equal to 0.986893274 and -0.986893274.Subproblem 1
x + 0.375 = 0.986893274 Simplifying x + 0.375 = 0.986893274 Reorder the terms: 0.375 + x = 0.986893274 Solving 0.375 + x = 0.986893274 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = 0.986893274 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 0.986893274 + -0.375 x = 0.986893274 + -0.375 Combine like terms: 0.986893274 + -0.375 = 0.611893274 x = 0.611893274 Simplifying x = 0.611893274Subproblem 2
x + 0.375 = -0.986893274 Simplifying x + 0.375 = -0.986893274 Reorder the terms: 0.375 + x = -0.986893274 Solving 0.375 + x = -0.986893274 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = -0.986893274 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -0.986893274 + -0.375 x = -0.986893274 + -0.375 Combine like terms: -0.986893274 + -0.375 = -1.361893274 x = -1.361893274 Simplifying x = -1.361893274Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.611893274, -1.361893274}
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