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Simplifying 0.5(8x2) + 0.5(14x(x + 0.5)) = 24 Remove parenthesis around (8x2) 0.5 * 8x2 + 0.5(14x(x + 0.5)) = 24 Multiply 0.5 * 8 4x2 + 0.5(14x(x + 0.5)) = 24 Reorder the terms: 4x2 + 0.5(14x(0.5 + x)) = 24 4x2 + 0.5((0.5 * 14x + x * 14x)) = 24 4x2 + 0.5((7x + 14x2)) = 24 4x2 + (7x * 0.5 + 14x2 * 0.5) = 24 4x2 + (3.5x + 7x2) = 24 Reorder the terms: 3.5x + 4x2 + 7x2 = 24 Combine like terms: 4x2 + 7x2 = 11x2 3.5x + 11x2 = 24 Solving 3.5x + 11x2 = 24 Solving for variable 'x'. Reorder the terms: -24 + 3.5x + 11x2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 3.5x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.181818182 + 0.3181818182x + x2 = 0 Move the constant term to the right: Add '2.181818182' to each side of the equation. -2.181818182 + 0.3181818182x + 2.181818182 + x2 = 0 + 2.181818182 Reorder the terms: -2.181818182 + 2.181818182 + 0.3181818182x + x2 = 0 + 2.181818182 Combine like terms: -2.181818182 + 2.181818182 = 0.000000000 0.000000000 + 0.3181818182x + x2 = 0 + 2.181818182 0.3181818182x + x2 = 0 + 2.181818182 Combine like terms: 0 + 2.181818182 = 2.181818182 0.3181818182x + x2 = 2.181818182 The x term is 0.3181818182x. Take half its coefficient (0.1590909091). Square it (0.02530991736) and add it to both sides. Add '0.02530991736' to each side of the equation. 0.3181818182x + 0.02530991736 + x2 = 2.181818182 + 0.02530991736 Reorder the terms: 0.02530991736 + 0.3181818182x + x2 = 2.181818182 + 0.02530991736 Combine like terms: 2.181818182 + 0.02530991736 = 2.20712809936 0.02530991736 + 0.3181818182x + x2 = 2.20712809936 Factor a perfect square on the left side: (x + 0.1590909091)(x + 0.1590909091) = 2.20712809936 Calculate the square root of the right side: 1.485640636 Break this problem into two subproblems by setting (x + 0.1590909091) equal to 1.485640636 and -1.485640636.Subproblem 1
x + 0.1590909091 = 1.485640636 Simplifying x + 0.1590909091 = 1.485640636 Reorder the terms: 0.1590909091 + x = 1.485640636 Solving 0.1590909091 + x = 1.485640636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + x = 1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + x = 1.485640636 + -0.1590909091 x = 1.485640636 + -0.1590909091 Combine like terms: 1.485640636 + -0.1590909091 = 1.3265497269 x = 1.3265497269 Simplifying x = 1.3265497269Subproblem 2
x + 0.1590909091 = -1.485640636 Simplifying x + 0.1590909091 = -1.485640636 Reorder the terms: 0.1590909091 + x = -1.485640636 Solving 0.1590909091 + x = -1.485640636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + x = -1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + x = -1.485640636 + -0.1590909091 x = -1.485640636 + -0.1590909091 Combine like terms: -1.485640636 + -0.1590909091 = -1.6447315451 x = -1.6447315451 Simplifying x = -1.6447315451Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3265497269, -1.6447315451}
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