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Simplifying 4x(3x + 5) = 2 Reorder the terms: 4x(5 + 3x) = 2 (5 * 4x + 3x * 4x) = 2 (20x + 12x2) = 2 Solving 20x + 12x2 = 2 Solving for variable 'x'. Reorder the terms: -2 + 20x + 12x2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + 20x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 10x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 10x + 6x2)' equal to zero and attempt to solve: Simplifying -1 + 10x + 6x2 = 0 Solving -1 + 10x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.1666666667 + 1.666666667x + x2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 1.666666667x + 0.1666666667 + x2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 1.666666667x + x2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 1.666666667x + x2 = 0 + 0.1666666667 1.666666667x + x2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 1.666666667x + x2 = 0.1666666667 The x term is 1.666666667x. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667x + 0.6944444447 + x2 = 0.1666666667 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667x + x2 = 0.1666666667 + 0.6944444447 Combine like terms: 0.1666666667 + 0.6944444447 = 0.8611111114 0.6944444447 + 1.666666667x + x2 = 0.8611111114 Factor a perfect square on the left side: (x + 0.8333333335)(x + 0.8333333335) = 0.8611111114 Calculate the square root of the right side: 0.927960727 Break this problem into two subproblems by setting (x + 0.8333333335) equal to 0.927960727 and -0.927960727.Subproblem 1
x + 0.8333333335 = 0.927960727 Simplifying x + 0.8333333335 = 0.927960727 Reorder the terms: 0.8333333335 + x = 0.927960727 Solving 0.8333333335 + x = 0.927960727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = 0.927960727 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 0.927960727 + -0.8333333335 x = 0.927960727 + -0.8333333335 Combine like terms: 0.927960727 + -0.8333333335 = 0.0946273935 x = 0.0946273935 Simplifying x = 0.0946273935Subproblem 2
x + 0.8333333335 = -0.927960727 Simplifying x + 0.8333333335 = -0.927960727 Reorder the terms: 0.8333333335 + x = -0.927960727 Solving 0.8333333335 + x = -0.927960727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = -0.927960727 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -0.927960727 + -0.8333333335 x = -0.927960727 + -0.8333333335 Combine like terms: -0.927960727 + -0.8333333335 = -1.7612940605 x = -1.7612940605 Simplifying x = -1.7612940605Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.0946273935, -1.7612940605}Solution
x = {0.0946273935, -1.7612940605}
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