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Simplifying 96 = (2x + 1)(x + 1) Reorder the terms: 96 = (1 + 2x)(x + 1) Reorder the terms: 96 = (1 + 2x)(1 + x) Multiply (1 + 2x) * (1 + x) 96 = (1(1 + x) + 2x * (1 + x)) 96 = ((1 * 1 + x * 1) + 2x * (1 + x)) 96 = ((1 + 1x) + 2x * (1 + x)) 96 = (1 + 1x + (1 * 2x + x * 2x)) 96 = (1 + 1x + (2x + 2x2)) Combine like terms: 1x + 2x = 3x 96 = (1 + 3x + 2x2) Solving 96 = 1 + 3x + 2x2 Solving for variable 'x'. Combine like terms: 96 + -1 = 95 95 + -3x + -2x2 = 1 + 3x + 2x2 + -1 + -3x + -2x2 Reorder the terms: 95 + -3x + -2x2 = 1 + -1 + 3x + -3x + 2x2 + -2x2 Combine like terms: 1 + -1 = 0 95 + -3x + -2x2 = 0 + 3x + -3x + 2x2 + -2x2 95 + -3x + -2x2 = 3x + -3x + 2x2 + -2x2 Combine like terms: 3x + -3x = 0 95 + -3x + -2x2 = 0 + 2x2 + -2x2 95 + -3x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 95 + -3x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -47.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '47.5' to each side of the equation. -47.5 + 1.5x + 47.5 + x2 = 0 + 47.5 Reorder the terms: -47.5 + 47.5 + 1.5x + x2 = 0 + 47.5 Combine like terms: -47.5 + 47.5 = 0.0 0.0 + 1.5x + x2 = 0 + 47.5 1.5x + x2 = 0 + 47.5 Combine like terms: 0 + 47.5 = 47.5 1.5x + x2 = 47.5 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 = 47.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 47.5 + 0.5625 Combine like terms: 47.5 + 0.5625 = 48.0625 0.5625 + 1.5x + x2 = 48.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 48.0625 Calculate the square root of the right side: 6.932712312 Break this problem into two subproblems by setting (x + 0.75) equal to 6.932712312 and -6.932712312.Subproblem 1
x + 0.75 = 6.932712312 Simplifying x + 0.75 = 6.932712312 Reorder the terms: 0.75 + x = 6.932712312 Solving 0.75 + x = 6.932712312 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 = 6.932712312 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 6.932712312 + -0.75 x = 6.932712312 + -0.75 Combine like terms: 6.932712312 + -0.75 = 6.182712312 x = 6.182712312 Simplifying x = 6.182712312Subproblem 2
x + 0.75 = -6.932712312 Simplifying x + 0.75 = -6.932712312 Reorder the terms: 0.75 + x = -6.932712312 Solving 0.75 + x = -6.932712312 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 = -6.932712312 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -6.932712312 + -0.75 x = -6.932712312 + -0.75 Combine like terms: -6.932712312 + -0.75 = -7.682712312 x = -7.682712312 Simplifying x = -7.682712312Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.182712312, -7.682712312}
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