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Simplifying x(2x + 3) = 190 Reorder the terms: x(3 + 2x) = 190 (3 * x + 2x * x) = 190 (3x + 2x2) = 190 Solving 3x + 2x2 = 190 Solving for variable 'x'. Reorder the terms: -190 + 3x + 2x2 = 190 + -190 Combine like terms: 190 + -190 = 0 -190 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -95 + 1.5x + x2 = 0 Move the constant term to the right: Add '95' to each side of the equation. -95 + 1.5x + 95 + x2 = 0 + 95 Reorder the terms: -95 + 95 + 1.5x + x2 = 0 + 95 Combine like terms: -95 + 95 = 0 0 + 1.5x + x2 = 0 + 95 1.5x + x2 = 0 + 95 Combine like terms: 0 + 95 = 95 1.5x + x2 = 95 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 95 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 95 + 0.5625 Combine like terms: 95 + 0.5625 = 95.5625 0.5625 + 1.5x + x2 = 95.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 95.5625 Calculate the square root of the right side: 9.775607398 Break this problem into two subproblems by setting (x + 0.75) equal to 9.775607398 and -9.775607398.Subproblem 1
x + 0.75 = 9.775607398 Simplifying x + 0.75 = 9.775607398 Reorder the terms: 0.75 + x = 9.775607398 Solving 0.75 + x = 9.775607398 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 9.775607398 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 9.775607398 + -0.75 x = 9.775607398 + -0.75 Combine like terms: 9.775607398 + -0.75 = 9.025607398 x = 9.025607398 Simplifying x = 9.025607398Subproblem 2
x + 0.75 = -9.775607398 Simplifying x + 0.75 = -9.775607398 Reorder the terms: 0.75 + x = -9.775607398 Solving 0.75 + x = -9.775607398 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -9.775607398 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -9.775607398 + -0.75 x = -9.775607398 + -0.75 Combine like terms: -9.775607398 + -0.75 = -10.525607398 x = -10.525607398 Simplifying x = -10.525607398Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.025607398, -10.525607398}
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