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Simplifying 180 = 4x2 + 2x Reorder the terms: 180 = 2x + 4x2 Solving 180 = 2x + 4x2 Solving for variable 'x'. Reorder the terms: 180 + -2x + -4x2 = 2x + -2x + 4x2 + -4x2 Combine like terms: 2x + -2x = 0 180 + -2x + -4x2 = 0 + 4x2 + -4x2 180 + -2x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 180 + -2x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(90 + -1x + -2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + -1x + -2x2)' equal to zero and attempt to solve: Simplifying 90 + -1x + -2x2 = 0 Solving 90 + -1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -45 + 0.5x + x2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 0.5x + 45 + x2 = 0 + 45 Reorder the terms: -45 + 45 + 0.5x + x2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 0.5x + x2 = 0 + 45 0.5x + x2 = 0 + 45 Combine like terms: 0 + 45 = 45 0.5x + x2 = 45 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 = 45 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 45 + 0.0625 Combine like terms: 45 + 0.0625 = 45.0625 0.0625 + 0.5x + x2 = 45.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 45.0625 Calculate the square root of the right side: 6.712860791 Break this problem into two subproblems by setting (x + 0.25) equal to 6.712860791 and -6.712860791.Subproblem 1
x + 0.25 = 6.712860791 Simplifying x + 0.25 = 6.712860791 Reorder the terms: 0.25 + x = 6.712860791 Solving 0.25 + x = 6.712860791 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 = 6.712860791 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 6.712860791 + -0.25 x = 6.712860791 + -0.25 Combine like terms: 6.712860791 + -0.25 = 6.462860791 x = 6.462860791 Simplifying x = 6.462860791Subproblem 2
x + 0.25 = -6.712860791 Simplifying x + 0.25 = -6.712860791 Reorder the terms: 0.25 + x = -6.712860791 Solving 0.25 + x = -6.712860791 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 = -6.712860791 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -6.712860791 + -0.25 x = -6.712860791 + -0.25 Combine like terms: -6.712860791 + -0.25 = -6.962860791 x = -6.962860791 Simplifying x = -6.962860791Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.462860791, -6.962860791}Solution
x = {6.462860791, -6.962860791}
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