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Simplifying x(2x + 1) = 63 Reorder the terms: x(1 + 2x) = 63 (1 * x + 2x * x) = 63 (1x + 2x2) = 63 Solving 1x + 2x2 = 63 Solving for variable 'x'. Reorder the terms: -63 + 1x + 2x2 = 63 + -63 Combine like terms: 63 + -63 = 0 -63 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -31.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '31.5' to each side of the equation. -31.5 + 0.5x + 31.5 + x2 = 0 + 31.5 Reorder the terms: -31.5 + 31.5 + 0.5x + x2 = 0 + 31.5 Combine like terms: -31.5 + 31.5 = 0.0 0.0 + 0.5x + x2 = 0 + 31.5 0.5x + x2 = 0 + 31.5 Combine like terms: 0 + 31.5 = 31.5 0.5x + x2 = 31.5 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 = 31.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 31.5 + 0.0625 Combine like terms: 31.5 + 0.0625 = 31.5625 0.0625 + 0.5x + x2 = 31.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 31.5625 Calculate the square root of the right side: 5.618051264 Break this problem into two subproblems by setting (x + 0.25) equal to 5.618051264 and -5.618051264.Subproblem 1
x + 0.25 = 5.618051264 Simplifying x + 0.25 = 5.618051264 Reorder the terms: 0.25 + x = 5.618051264 Solving 0.25 + x = 5.618051264 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 = 5.618051264 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.618051264 + -0.25 x = 5.618051264 + -0.25 Combine like terms: 5.618051264 + -0.25 = 5.368051264 x = 5.368051264 Simplifying x = 5.368051264Subproblem 2
x + 0.25 = -5.618051264 Simplifying x + 0.25 = -5.618051264 Reorder the terms: 0.25 + x = -5.618051264 Solving 0.25 + x = -5.618051264 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 = -5.618051264 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.618051264 + -0.25 x = -5.618051264 + -0.25 Combine like terms: -5.618051264 + -0.25 = -5.868051264 x = -5.868051264 Simplifying x = -5.868051264Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.368051264, -5.868051264}
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