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Simplifying 0.5(2x + 9) * x = 100 Reorder the terms: 0.5(9 + 2x) * x = 100 Reorder the terms for easier multiplication: 0.5x(9 + 2x) = 100 (9 * 0.5x + 2x * 0.5x) = 100 (4.5x + 1x2) = 100 Solving 4.5x + 1x2 = 100 Solving for variable 'x'. Reorder the terms: -100 + 4.5x + 1x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -100 + 4.5x + 1x2 = 0 Begin completing the square. Move the constant term to the right: Add '100' to each side of the equation. -100 + 4.5x + 100 + x2 = 0 + 100 Reorder the terms: -100 + 100 + 4.5x + x2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + 4.5x + x2 = 0 + 100 4.5x + x2 = 0 + 100 Combine like terms: 0 + 100 = 100 4.5x + x2 = 100 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 100 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 100 + 5.0625 Combine like terms: 100 + 5.0625 = 105.0625 5.0625 + 4.5x + x2 = 105.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 105.0625 Calculate the square root of the right side: 10.25 Break this problem into two subproblems by setting (x + 2.25) equal to 10.25 and -10.25.Subproblem 1
x + 2.25 = 10.25 Simplifying x + 2.25 = 10.25 Reorder the terms: 2.25 + x = 10.25 Solving 2.25 + x = 10.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 10.25 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 10.25 + -2.25 x = 10.25 + -2.25 Combine like terms: 10.25 + -2.25 = 8 x = 8 Simplifying x = 8Subproblem 2
x + 2.25 = -10.25 Simplifying x + 2.25 = -10.25 Reorder the terms: 2.25 + x = -10.25 Solving 2.25 + x = -10.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -10.25 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -10.25 + -2.25 x = -10.25 + -2.25 Combine like terms: -10.25 + -2.25 = -12.5 x = -12.5 Simplifying x = -12.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {8, -12.5}
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