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Simplifying (2x + -1)(2x + -1) = 7 Reorder the terms: (-1 + 2x)(2x + -1) = 7 Reorder the terms: (-1 + 2x)(-1 + 2x) = 7 Multiply (-1 + 2x) * (-1 + 2x) (-1(-1 + 2x) + 2x * (-1 + 2x)) = 7 ((-1 * -1 + 2x * -1) + 2x * (-1 + 2x)) = 7 ((1 + -2x) + 2x * (-1 + 2x)) = 7 (1 + -2x + (-1 * 2x + 2x * 2x)) = 7 (1 + -2x + (-2x + 4x2)) = 7 Combine like terms: -2x + -2x = -4x (1 + -4x + 4x2) = 7 Solving 1 + -4x + 4x2 = 7 Solving for variable 'x'. Reorder the terms: 1 + -7 + -4x + 4x2 = 7 + -7 Combine like terms: 1 + -7 = -6 -6 + -4x + 4x2 = 7 + -7 Combine like terms: 7 + -7 = 0 -6 + -4x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -2x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -2x + 2x2)' equal to zero and attempt to solve: Simplifying -3 + -2x + 2x2 = 0 Solving -3 + -2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + -1x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1x + x2 = 0 + 1.5 -1x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1x + x2 = 1.5 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 1.5 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 0.25 + -1x + x2 = 1.75 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (x + -0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
x + -0.5 = 1.322875656 Simplifying x + -0.5 = 1.322875656 Reorder the terms: -0.5 + x = 1.322875656 Solving -0.5 + x = 1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.322875656 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.322875656 + 0.5 x = 1.322875656 + 0.5 Combine like terms: 1.322875656 + 0.5 = 1.822875656 x = 1.822875656 Simplifying x = 1.822875656Subproblem 2
x + -0.5 = -1.322875656 Simplifying x + -0.5 = -1.322875656 Reorder the terms: -0.5 + x = -1.322875656 Solving -0.5 + x = -1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.322875656 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.322875656 + 0.5 x = -1.322875656 + 0.5 Combine like terms: -1.322875656 + 0.5 = -0.822875656 x = -0.822875656 Simplifying x = -0.822875656Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.822875656, -0.822875656}Solution
x = {1.822875656, -0.822875656}
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