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Simplifying 7x(1 + x) + 8(-6x + -5) = 0 (1 * 7x + x * 7x) + 8(-6x + -5) = 0 (7x + 7x2) + 8(-6x + -5) = 0 Reorder the terms: 7x + 7x2 + 8(-5 + -6x) = 0 7x + 7x2 + (-5 * 8 + -6x * 8) = 0 7x + 7x2 + (-40 + -48x) = 0 Reorder the terms: -40 + 7x + -48x + 7x2 = 0 Combine like terms: 7x + -48x = -41x -40 + -41x + 7x2 = 0 Solving -40 + -41x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5.714285714 + -5.857142857x + x2 = 0 Move the constant term to the right: Add '5.714285714' to each side of the equation. -5.714285714 + -5.857142857x + 5.714285714 + x2 = 0 + 5.714285714 Reorder the terms: -5.714285714 + 5.714285714 + -5.857142857x + x2 = 0 + 5.714285714 Combine like terms: -5.714285714 + 5.714285714 = 0.000000000 0.000000000 + -5.857142857x + x2 = 0 + 5.714285714 -5.857142857x + x2 = 0 + 5.714285714 Combine like terms: 0 + 5.714285714 = 5.714285714 -5.857142857x + x2 = 5.714285714 The x term is -5.857142857x. Take half its coefficient (-2.928571429). Square it (8.576530615) and add it to both sides. Add '8.576530615' to each side of the equation. -5.857142857x + 8.576530615 + x2 = 5.714285714 + 8.576530615 Reorder the terms: 8.576530615 + -5.857142857x + x2 = 5.714285714 + 8.576530615 Combine like terms: 5.714285714 + 8.576530615 = 14.290816329 8.576530615 + -5.857142857x + x2 = 14.290816329 Factor a perfect square on the left side: (x + -2.928571429)(x + -2.928571429) = 14.290816329 Calculate the square root of the right side: 3.780319607 Break this problem into two subproblems by setting (x + -2.928571429) equal to 3.780319607 and -3.780319607.Subproblem 1
x + -2.928571429 = 3.780319607 Simplifying x + -2.928571429 = 3.780319607 Reorder the terms: -2.928571429 + x = 3.780319607 Solving -2.928571429 + x = 3.780319607 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + x = 3.780319607 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + x = 3.780319607 + 2.928571429 x = 3.780319607 + 2.928571429 Combine like terms: 3.780319607 + 2.928571429 = 6.708891036 x = 6.708891036 Simplifying x = 6.708891036Subproblem 2
x + -2.928571429 = -3.780319607 Simplifying x + -2.928571429 = -3.780319607 Reorder the terms: -2.928571429 + x = -3.780319607 Solving -2.928571429 + x = -3.780319607 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + x = -3.780319607 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + x = -3.780319607 + 2.928571429 x = -3.780319607 + 2.928571429 Combine like terms: -3.780319607 + 2.928571429 = -0.851748178 x = -0.851748178 Simplifying x = -0.851748178Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.708891036, -0.851748178}
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