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Simplifying 0.5(x)(2x + 1) = 18 Reorder the terms: 0.5x(1 + 2x) = 18 (1 * 0.5x + 2x * 0.5x) = 18 (0.5x + 1x2) = 18 Solving 0.5x + 1x2 = 18 Solving for variable 'x'. Reorder the terms: -18 + 0.5x + 1x2 = 18 + -18 Combine like terms: 18 + -18 = 0 -18 + 0.5x + 1x2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 0.5x + 18 + x2 = 0 + 18 Reorder the terms: -18 + 18 + 0.5x + x2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 0.5x + x2 = 0 + 18 0.5x + x2 = 0 + 18 Combine like terms: 0 + 18 = 18 0.5x + x2 = 18 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 18 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 18 + 0.0625 Combine like terms: 18 + 0.0625 = 18.0625 0.0625 + 0.5x + x2 = 18.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 18.0625 Calculate the square root of the right side: 4.25 Break this problem into two subproblems by setting (x + 0.25) equal to 4.25 and -4.25.Subproblem 1
x + 0.25 = 4.25 Simplifying x + 0.25 = 4.25 Reorder the terms: 0.25 + x = 4.25 Solving 0.25 + x = 4.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 4.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 4.25 + -0.25 x = 4.25 + -0.25 Combine like terms: 4.25 + -0.25 = 4 x = 4 Simplifying x = 4Subproblem 2
x + 0.25 = -4.25 Simplifying x + 0.25 = -4.25 Reorder the terms: 0.25 + x = -4.25 Solving 0.25 + x = -4.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -4.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -4.25 + -0.25 x = -4.25 + -0.25 Combine like terms: -4.25 + -0.25 = -4.5 x = -4.5 Simplifying x = -4.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {4, -4.5}
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