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Simplifying 0 = 15x2 + 22.5x + -405 Reorder the terms: 0 = -405 + 22.5x + 15x2 Solving 0 = -405 + 22.5x + 15x2 Solving for variable 'x'. Combine like terms: 0 + 405 = 405 405 + -22.5x + -15x2 = -405 + 22.5x + 15x2 + 405 + -22.5x + -15x2 Reorder the terms: 405 + -22.5x + -15x2 = -405 + 405 + 22.5x + -22.5x + 15x2 + -15x2 Combine like terms: -405 + 405 = 0 405 + -22.5x + -15x2 = 0 + 22.5x + -22.5x + 15x2 + -15x2 405 + -22.5x + -15x2 = 22.5x + -22.5x + 15x2 + -15x2 Combine like terms: 22.5x + -22.5x = 0.0 405 + -22.5x + -15x2 = 0.0 + 15x2 + -15x2 405 + -22.5x + -15x2 = 15x2 + -15x2 Combine like terms: 15x2 + -15x2 = 0 405 + -22.5x + -15x2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -27 + 1.5x + x2 = 0 Move the constant term to the right: Add '27' to each side of the equation. -27 + 1.5x + 27 + x2 = 0 + 27 Reorder the terms: -27 + 27 + 1.5x + x2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + 1.5x + x2 = 0 + 27 1.5x + x2 = 0 + 27 Combine like terms: 0 + 27 = 27 1.5x + x2 = 27 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 = 27 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 27 + 0.5625 Combine like terms: 27 + 0.5625 = 27.5625 0.5625 + 1.5x + x2 = 27.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 27.5625 Calculate the square root of the right side: 5.25 Break this problem into two subproblems by setting (x + 0.75) equal to 5.25 and -5.25.Subproblem 1
x + 0.75 = 5.25 Simplifying x + 0.75 = 5.25 Reorder the terms: 0.75 + x = 5.25 Solving 0.75 + x = 5.25 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 = 5.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 5.25 + -0.75 x = 5.25 + -0.75 Combine like terms: 5.25 + -0.75 = 4.5 x = 4.5 Simplifying x = 4.5Subproblem 2
x + 0.75 = -5.25 Simplifying x + 0.75 = -5.25 Reorder the terms: 0.75 + x = -5.25 Solving 0.75 + x = -5.25 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 = -5.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -5.25 + -0.75 x = -5.25 + -0.75 Combine like terms: -5.25 + -0.75 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.5, -6}
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