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Simplifying 4x2 + 6x = 1680 Reorder the terms: 6x + 4x2 = 1680 Solving 6x + 4x2 = 1680 Solving for variable 'x'. Reorder the terms: -1680 + 6x + 4x2 = 1680 + -1680 Combine like terms: 1680 + -1680 = 0 -1680 + 6x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-840 + 3x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-840 + 3x + 2x2)' equal to zero and attempt to solve: Simplifying -840 + 3x + 2x2 = 0 Solving -840 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -420 + 1.5x + x2 = 0 Move the constant term to the right: Add '420' to each side of the equation. -420 + 1.5x + 420 + x2 = 0 + 420 Reorder the terms: -420 + 420 + 1.5x + x2 = 0 + 420 Combine like terms: -420 + 420 = 0 0 + 1.5x + x2 = 0 + 420 1.5x + x2 = 0 + 420 Combine like terms: 0 + 420 = 420 1.5x + x2 = 420 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 = 420 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 420 + 0.5625 Combine like terms: 420 + 0.5625 = 420.5625 0.5625 + 1.5x + x2 = 420.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 420.5625 Calculate the square root of the right side: 20.507620535 Break this problem into two subproblems by setting (x + 0.75) equal to 20.507620535 and -20.507620535.Subproblem 1
x + 0.75 = 20.507620535 Simplifying x + 0.75 = 20.507620535 Reorder the terms: 0.75 + x = 20.507620535 Solving 0.75 + x = 20.507620535 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 = 20.507620535 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 20.507620535 + -0.75 x = 20.507620535 + -0.75 Combine like terms: 20.507620535 + -0.75 = 19.757620535 x = 19.757620535 Simplifying x = 19.757620535Subproblem 2
x + 0.75 = -20.507620535 Simplifying x + 0.75 = -20.507620535 Reorder the terms: 0.75 + x = -20.507620535 Solving 0.75 + x = -20.507620535 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 = -20.507620535 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -20.507620535 + -0.75 x = -20.507620535 + -0.75 Combine like terms: -20.507620535 + -0.75 = -21.257620535 x = -21.257620535 Simplifying x = -21.257620535Solution
The solution to the problem is based on the solutions from the subproblems. x = {19.757620535, -21.257620535}Solution
x = {19.757620535, -21.257620535}
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