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Simplifying 91 = (2x + -3)(x + 2) Reorder the terms: 91 = (-3 + 2x)(x + 2) Reorder the terms: 91 = (-3 + 2x)(2 + x) Multiply (-3 + 2x) * (2 + x) 91 = (-3(2 + x) + 2x * (2 + x)) 91 = ((2 * -3 + x * -3) + 2x * (2 + x)) 91 = ((-6 + -3x) + 2x * (2 + x)) 91 = (-6 + -3x + (2 * 2x + x * 2x)) 91 = (-6 + -3x + (4x + 2x2)) Combine like terms: -3x + 4x = 1x 91 = (-6 + 1x + 2x2) Solving 91 = -6 + 1x + 2x2 Solving for variable 'x'. Combine like terms: 91 + 6 = 97 97 + -1x + -2x2 = -6 + 1x + 2x2 + 6 + -1x + -2x2 Reorder the terms: 97 + -1x + -2x2 = -6 + 6 + 1x + -1x + 2x2 + -2x2 Combine like terms: -6 + 6 = 0 97 + -1x + -2x2 = 0 + 1x + -1x + 2x2 + -2x2 97 + -1x + -2x2 = 1x + -1x + 2x2 + -2x2 Combine like terms: 1x + -1x = 0 97 + -1x + -2x2 = 0 + 2x2 + -2x2 97 + -1x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 97 + -1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -48.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '48.5' to each side of the equation. -48.5 + 0.5x + 48.5 + x2 = 0 + 48.5 Reorder the terms: -48.5 + 48.5 + 0.5x + x2 = 0 + 48.5 Combine like terms: -48.5 + 48.5 = 0.0 0.0 + 0.5x + x2 = 0 + 48.5 0.5x + x2 = 0 + 48.5 Combine like terms: 0 + 48.5 = 48.5 0.5x + x2 = 48.5 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 = 48.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 48.5 + 0.0625 Combine like terms: 48.5 + 0.0625 = 48.5625 0.0625 + 0.5x + x2 = 48.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 48.5625 Calculate the square root of the right side: 6.968679932 Break this problem into two subproblems by setting (x + 0.25) equal to 6.968679932 and -6.968679932.Subproblem 1
x + 0.25 = 6.968679932 Simplifying x + 0.25 = 6.968679932 Reorder the terms: 0.25 + x = 6.968679932 Solving 0.25 + x = 6.968679932 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 = 6.968679932 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 6.968679932 + -0.25 x = 6.968679932 + -0.25 Combine like terms: 6.968679932 + -0.25 = 6.718679932 x = 6.718679932 Simplifying x = 6.718679932Subproblem 2
x + 0.25 = -6.968679932 Simplifying x + 0.25 = -6.968679932 Reorder the terms: 0.25 + x = -6.968679932 Solving 0.25 + x = -6.968679932 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 = -6.968679932 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -6.968679932 + -0.25 x = -6.968679932 + -0.25 Combine like terms: -6.968679932 + -0.25 = -7.218679932 x = -7.218679932 Simplifying x = -7.218679932Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.718679932, -7.218679932}
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