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Simplifying 1 + 1(3x + 5) = 4(3x + 5)(4x + -5) Reorder the terms: 1 + 1(5 + 3x) = 4(3x + 5)(4x + -5) 1 + (5 * 1 + 3x * 1) = 4(3x + 5)(4x + -5) 1 + (5 + 3x) = 4(3x + 5)(4x + -5) Combine like terms: 1 + 5 = 6 6 + 3x = 4(3x + 5)(4x + -5) Reorder the terms: 6 + 3x = 4(5 + 3x)(4x + -5) Reorder the terms: 6 + 3x = 4(5 + 3x)(-5 + 4x) Multiply (5 + 3x) * (-5 + 4x) 6 + 3x = 4(5(-5 + 4x) + 3x * (-5 + 4x)) 6 + 3x = 4((-5 * 5 + 4x * 5) + 3x * (-5 + 4x)) 6 + 3x = 4((-25 + 20x) + 3x * (-5 + 4x)) 6 + 3x = 4(-25 + 20x + (-5 * 3x + 4x * 3x)) 6 + 3x = 4(-25 + 20x + (-15x + 12x2)) Combine like terms: 20x + -15x = 5x 6 + 3x = 4(-25 + 5x + 12x2) 6 + 3x = (-25 * 4 + 5x * 4 + 12x2 * 4) 6 + 3x = (-100 + 20x + 48x2) Solving 6 + 3x = -100 + 20x + 48x2 Solving for variable 'x'. Reorder the terms: 6 + 100 + 3x + -20x + -48x2 = -100 + 20x + 48x2 + 100 + -20x + -48x2 Combine like terms: 6 + 100 = 106 106 + 3x + -20x + -48x2 = -100 + 20x + 48x2 + 100 + -20x + -48x2 Combine like terms: 3x + -20x = -17x 106 + -17x + -48x2 = -100 + 20x + 48x2 + 100 + -20x + -48x2 Reorder the terms: 106 + -17x + -48x2 = -100 + 100 + 20x + -20x + 48x2 + -48x2 Combine like terms: -100 + 100 = 0 106 + -17x + -48x2 = 0 + 20x + -20x + 48x2 + -48x2 106 + -17x + -48x2 = 20x + -20x + 48x2 + -48x2 Combine like terms: 20x + -20x = 0 106 + -17x + -48x2 = 0 + 48x2 + -48x2 106 + -17x + -48x2 = 48x2 + -48x2 Combine like terms: 48x2 + -48x2 = 0 106 + -17x + -48x2 = 0 Begin completing the square. Divide all terms by -48 the coefficient of the squared term: Divide each side by '-48'. -2.208333333 + 0.3541666667x + x2 = 0 Move the constant term to the right: Add '2.208333333' to each side of the equation. -2.208333333 + 0.3541666667x + 2.208333333 + x2 = 0 + 2.208333333 Reorder the terms: -2.208333333 + 2.208333333 + 0.3541666667x + x2 = 0 + 2.208333333 Combine like terms: -2.208333333 + 2.208333333 = 0.000000000 0.000000000 + 0.3541666667x + x2 = 0 + 2.208333333 0.3541666667x + x2 = 0 + 2.208333333 Combine like terms: 0 + 2.208333333 = 2.208333333 0.3541666667x + x2 = 2.208333333 The x term is 0.3541666667x. Take half its coefficient (0.1770833334). Square it (0.03135850697) and add it to both sides. Add '0.03135850697' to each side of the equation. 0.3541666667x + 0.03135850697 + x2 = 2.208333333 + 0.03135850697 Reorder the terms: 0.03135850697 + 0.3541666667x + x2 = 2.208333333 + 0.03135850697 Combine like terms: 2.208333333 + 0.03135850697 = 2.23969183997 0.03135850697 + 0.3541666667x + x2 = 2.23969183997 Factor a perfect square on the left side: (x + 0.1770833334)(x + 0.1770833334) = 2.23969183997 Calculate the square root of the right side: 1.496560002 Break this problem into two subproblems by setting (x + 0.1770833334) equal to 1.496560002 and -1.496560002.Subproblem 1
x + 0.1770833334 = 1.496560002 Simplifying x + 0.1770833334 = 1.496560002 Reorder the terms: 0.1770833334 + x = 1.496560002 Solving 0.1770833334 + x = 1.496560002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1770833334' to each side of the equation. 0.1770833334 + -0.1770833334 + x = 1.496560002 + -0.1770833334 Combine like terms: 0.1770833334 + -0.1770833334 = 0.0000000000 0.0000000000 + x = 1.496560002 + -0.1770833334 x = 1.496560002 + -0.1770833334 Combine like terms: 1.496560002 + -0.1770833334 = 1.3194766686 x = 1.3194766686 Simplifying x = 1.3194766686Subproblem 2
x + 0.1770833334 = -1.496560002 Simplifying x + 0.1770833334 = -1.496560002 Reorder the terms: 0.1770833334 + x = -1.496560002 Solving 0.1770833334 + x = -1.496560002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1770833334' to each side of the equation. 0.1770833334 + -0.1770833334 + x = -1.496560002 + -0.1770833334 Combine like terms: 0.1770833334 + -0.1770833334 = 0.0000000000 0.0000000000 + x = -1.496560002 + -0.1770833334 x = -1.496560002 + -0.1770833334 Combine like terms: -1.496560002 + -0.1770833334 = -1.6736433354 x = -1.6736433354 Simplifying x = -1.6736433354Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3194766686, -1.6736433354}
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