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Simplifying -0.7x2 + 7.6x = 1.5x + 5.4 Reorder the terms: 7.6x + -0.7x2 = 1.5x + 5.4 Reorder the terms: 7.6x + -0.7x2 = 5.4 + 1.5x Solving 7.6x + -0.7x2 = 5.4 + 1.5x Solving for variable 'x'. Reorder the terms: -5.4 + 7.6x + -1.5x + -0.7x2 = 5.4 + 1.5x + -5.4 + -1.5x Combine like terms: 7.6x + -1.5x = 6.1x -5.4 + 6.1x + -0.7x2 = 5.4 + 1.5x + -5.4 + -1.5x Reorder the terms: -5.4 + 6.1x + -0.7x2 = 5.4 + -5.4 + 1.5x + -1.5x Combine like terms: 5.4 + -5.4 = 0.0 -5.4 + 6.1x + -0.7x2 = 0.0 + 1.5x + -1.5x -5.4 + 6.1x + -0.7x2 = 1.5x + -1.5x Combine like terms: 1.5x + -1.5x = 0.0 -5.4 + 6.1x + -0.7x2 = 0.0 Begin completing the square. Divide all terms by -0.7 the coefficient of the squared term: Divide each side by '-0.7'. 7.714285714 + -8.714285714x + x2 = 0 Move the constant term to the right: Add '-7.714285714' to each side of the equation. 7.714285714 + -8.714285714x + -7.714285714 + x2 = 0 + -7.714285714 Reorder the terms: 7.714285714 + -7.714285714 + -8.714285714x + x2 = 0 + -7.714285714 Combine like terms: 7.714285714 + -7.714285714 = 0.000000000 0.000000000 + -8.714285714x + x2 = 0 + -7.714285714 -8.714285714x + x2 = 0 + -7.714285714 Combine like terms: 0 + -7.714285714 = -7.714285714 -8.714285714x + x2 = -7.714285714 The x term is -8.714285714x. Take half its coefficient (-4.357142857). Square it (18.98469388) and add it to both sides. Add '18.98469388' to each side of the equation. -8.714285714x + 18.98469388 + x2 = -7.714285714 + 18.98469388 Reorder the terms: 18.98469388 + -8.714285714x + x2 = -7.714285714 + 18.98469388 Combine like terms: -7.714285714 + 18.98469388 = 11.270408166 18.98469388 + -8.714285714x + x2 = 11.270408166 Factor a perfect square on the left side: (x + -4.357142857)(x + -4.357142857) = 11.270408166 Calculate the square root of the right side: 3.357142858 Break this problem into two subproblems by setting (x + -4.357142857) equal to 3.357142858 and -3.357142858.Subproblem 1
x + -4.357142857 = 3.357142858 Simplifying x + -4.357142857 = 3.357142858 Reorder the terms: -4.357142857 + x = 3.357142858 Solving -4.357142857 + x = 3.357142858 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.357142857' to each side of the equation. -4.357142857 + 4.357142857 + x = 3.357142858 + 4.357142857 Combine like terms: -4.357142857 + 4.357142857 = 0.000000000 0.000000000 + x = 3.357142858 + 4.357142857 x = 3.357142858 + 4.357142857 Combine like terms: 3.357142858 + 4.357142857 = 7.714285715 x = 7.714285715 Simplifying x = 7.714285715Subproblem 2
x + -4.357142857 = -3.357142858 Simplifying x + -4.357142857 = -3.357142858 Reorder the terms: -4.357142857 + x = -3.357142858 Solving -4.357142857 + x = -3.357142858 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.357142857' to each side of the equation. -4.357142857 + 4.357142857 + x = -3.357142858 + 4.357142857 Combine like terms: -4.357142857 + 4.357142857 = 0.000000000 0.000000000 + x = -3.357142858 + 4.357142857 x = -3.357142858 + 4.357142857 Combine like terms: -3.357142858 + 4.357142857 = 0.999999999 x = 0.999999999 Simplifying x = 0.999999999Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.714285715, 0.999999999}
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