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Simplifying 23 = x + 2x(x + 3) Reorder the terms: 23 = x + 2x(3 + x) 23 = x + (3 * 2x + x * 2x) 23 = x + (6x + 2x2) Combine like terms: x + 6x = 7x 23 = 7x + 2x2 Solving 23 = 7x + 2x2 Solving for variable 'x'. Reorder the terms: 23 + -7x + -2x2 = 7x + -7x + 2x2 + -2x2 Combine like terms: 7x + -7x = 0 23 + -7x + -2x2 = 0 + 2x2 + -2x2 23 + -7x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 23 + -7x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -11.5 + 3.5x + x2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 3.5x + 11.5 + x2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 3.5x + x2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 3.5x + x2 = 0 + 11.5 3.5x + x2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 3.5x + x2 = 11.5 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = 11.5 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 11.5 + 3.0625 Combine like terms: 11.5 + 3.0625 = 14.5625 3.0625 + 3.5x + x2 = 14.5625 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 14.5625 Calculate the square root of the right side: 3.816084381 Break this problem into two subproblems by setting (x + 1.75) equal to 3.816084381 and -3.816084381.Subproblem 1
x + 1.75 = 3.816084381 Simplifying x + 1.75 = 3.816084381 Reorder the terms: 1.75 + x = 3.816084381 Solving 1.75 + x = 3.816084381 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 3.816084381 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 3.816084381 + -1.75 x = 3.816084381 + -1.75 Combine like terms: 3.816084381 + -1.75 = 2.066084381 x = 2.066084381 Simplifying x = 2.066084381Subproblem 2
x + 1.75 = -3.816084381 Simplifying x + 1.75 = -3.816084381 Reorder the terms: 1.75 + x = -3.816084381 Solving 1.75 + x = -3.816084381 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -3.816084381 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -3.816084381 + -1.75 x = -3.816084381 + -1.75 Combine like terms: -3.816084381 + -1.75 = -5.566084381 x = -5.566084381 Simplifying x = -5.566084381Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.066084381, -5.566084381}
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