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Simplifying 7.5(3x + 4) * x = 7 Reorder the terms: 7.5(4 + 3x) * x = 7 Reorder the terms for easier multiplication: 7.5x(4 + 3x) = 7 (4 * 7.5x + 3x * 7.5x) = 7 (30x + 22.5x2) = 7 Solving 30x + 22.5x2 = 7 Solving for variable 'x'. Reorder the terms: -7 + 30x + 22.5x2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 30x + 22.5x2 = 0 Begin completing the square. Divide all terms by 22.5 the coefficient of the squared term: Divide each side by '22.5'. -0.3111111111 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '0.3111111111' to each side of the equation. -0.3111111111 + 1.333333333x + 0.3111111111 + x2 = 0 + 0.3111111111 Reorder the terms: -0.3111111111 + 0.3111111111 + 1.333333333x + x2 = 0 + 0.3111111111 Combine like terms: -0.3111111111 + 0.3111111111 = 0.0000000000 0.0000000000 + 1.333333333x + x2 = 0 + 0.3111111111 1.333333333x + x2 = 0 + 0.3111111111 Combine like terms: 0 + 0.3111111111 = 0.3111111111 1.333333333x + x2 = 0.3111111111 The x term is 1.333333333x. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333x + 0.4444444442 + x2 = 0.3111111111 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = 0.3111111111 + 0.4444444442 Combine like terms: 0.3111111111 + 0.4444444442 = 0.7555555553 0.4444444442 + 1.333333333x + x2 = 0.7555555553 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 0.7555555553 Calculate the square root of the right side: 0.869226987 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 0.869226987 and -0.869226987.Subproblem 1
x + 0.6666666665 = 0.869226987 Simplifying x + 0.6666666665 = 0.869226987 Reorder the terms: 0.6666666665 + x = 0.869226987 Solving 0.6666666665 + x = 0.869226987 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 0.869226987 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 0.869226987 + -0.6666666665 x = 0.869226987 + -0.6666666665 Combine like terms: 0.869226987 + -0.6666666665 = 0.2025603205 x = 0.2025603205 Simplifying x = 0.2025603205Subproblem 2
x + 0.6666666665 = -0.869226987 Simplifying x + 0.6666666665 = -0.869226987 Reorder the terms: 0.6666666665 + x = -0.869226987 Solving 0.6666666665 + x = -0.869226987 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -0.869226987 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -0.869226987 + -0.6666666665 x = -0.869226987 + -0.6666666665 Combine like terms: -0.869226987 + -0.6666666665 = -1.5358936535 x = -1.5358936535 Simplifying x = -1.5358936535Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2025603205, -1.5358936535}
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