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Simplifying 10x + [25x(2x + 3)] + [5(2x + 3)] = 31 Reorder the terms: 10x + [25x(3 + 2x)] + [5(2x + 3)] = 31 10x + [(3 * 25x + 2x * 25x)] + [5(2x + 3)] = 31 10x + [(75x + 50x2)] + [5(2x + 3)] = 31 10x + [75x + 50x2] + [5(2x + 3)] = 31 Remove brackets around [75x + 50x2] 10x + 75x + 50x2 + [5(2x + 3)] = 31 Reorder the terms: 10x + 75x + 50x2 + [5(3 + 2x)] = 31 10x + 75x + 50x2 + [(3 * 5 + 2x * 5)] = 31 10x + 75x + 50x2 + [(15 + 10x)] = 31 10x + 75x + 50x2 + [15 + 10x] = 31 Remove brackets around [15 + 10x] 10x + 75x + 50x2 + 15 + 10x = 31 Reorder the terms: 15 + 10x + 75x + 10x + 50x2 = 31 Combine like terms: 10x + 75x = 85x 15 + 85x + 10x + 50x2 = 31 Combine like terms: 85x + 10x = 95x 15 + 95x + 50x2 = 31 Solving 15 + 95x + 50x2 = 31 Solving for variable 'x'. Reorder the terms: 15 + -31 + 95x + 50x2 = 31 + -31 Combine like terms: 15 + -31 = -16 -16 + 95x + 50x2 = 31 + -31 Combine like terms: 31 + -31 = 0 -16 + 95x + 50x2 = 0 Begin completing the square. Divide all terms by 50 the coefficient of the squared term: Divide each side by '50'. -0.32 + 1.9x + x2 = 0.0 Move the constant term to the right: Add '0.32' to each side of the equation. -0.32 + 1.9x + 0.32 + x2 = 0.0 + 0.32 Reorder the terms: -0.32 + 0.32 + 1.9x + x2 = 0.0 + 0.32 Combine like terms: -0.32 + 0.32 = 0.00 0.00 + 1.9x + x2 = 0.0 + 0.32 1.9x + x2 = 0.0 + 0.32 Combine like terms: 0.0 + 0.32 = 0.32 1.9x + x2 = 0.32 The x term is 1.9x. Take half its coefficient (0.95). Square it (0.9025) and add it to both sides. Add '0.9025' to each side of the equation. 1.9x + 0.9025 + x2 = 0.32 + 0.9025 Reorder the terms: 0.9025 + 1.9x + x2 = 0.32 + 0.9025 Combine like terms: 0.32 + 0.9025 = 1.2225 0.9025 + 1.9x + x2 = 1.2225 Factor a perfect square on the left side: (x + 0.95)(x + 0.95) = 1.2225 Calculate the square root of the right side: 1.105667219 Break this problem into two subproblems by setting (x + 0.95) equal to 1.105667219 and -1.105667219.Subproblem 1
x + 0.95 = 1.105667219 Simplifying x + 0.95 = 1.105667219 Reorder the terms: 0.95 + x = 1.105667219 Solving 0.95 + x = 1.105667219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.95' to each side of the equation. 0.95 + -0.95 + x = 1.105667219 + -0.95 Combine like terms: 0.95 + -0.95 = 0.00 0.00 + x = 1.105667219 + -0.95 x = 1.105667219 + -0.95 Combine like terms: 1.105667219 + -0.95 = 0.155667219 x = 0.155667219 Simplifying x = 0.155667219Subproblem 2
x + 0.95 = -1.105667219 Simplifying x + 0.95 = -1.105667219 Reorder the terms: 0.95 + x = -1.105667219 Solving 0.95 + x = -1.105667219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.95' to each side of the equation. 0.95 + -0.95 + x = -1.105667219 + -0.95 Combine like terms: 0.95 + -0.95 = 0.00 0.00 + x = -1.105667219 + -0.95 x = -1.105667219 + -0.95 Combine like terms: -1.105667219 + -0.95 = -2.055667219 x = -2.055667219 Simplifying x = -2.055667219Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.155667219, -2.055667219}
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