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Simplifying 0.5x2 + 10 = -3x2 + -6x + 86 Reorder the terms: 10 + 0.5x2 = -3x2 + -6x + 86 Reorder the terms: 10 + 0.5x2 = 86 + -6x + -3x2 Solving 10 + 0.5x2 = 86 + -6x + -3x2 Solving for variable 'x'. Reorder the terms: 10 + -86 + 6x + 0.5x2 + 3x2 = 86 + -6x + -3x2 + -86 + 6x + 3x2 Combine like terms: 10 + -86 = -76 -76 + 6x + 0.5x2 + 3x2 = 86 + -6x + -3x2 + -86 + 6x + 3x2 Combine like terms: 0.5x2 + 3x2 = 3.5x2 -76 + 6x + 3.5x2 = 86 + -6x + -3x2 + -86 + 6x + 3x2 Reorder the terms: -76 + 6x + 3.5x2 = 86 + -86 + -6x + 6x + -3x2 + 3x2 Combine like terms: 86 + -86 = 0 -76 + 6x + 3.5x2 = 0 + -6x + 6x + -3x2 + 3x2 -76 + 6x + 3.5x2 = -6x + 6x + -3x2 + 3x2 Combine like terms: -6x + 6x = 0 -76 + 6x + 3.5x2 = 0 + -3x2 + 3x2 -76 + 6x + 3.5x2 = -3x2 + 3x2 Combine like terms: -3x2 + 3x2 = 0 -76 + 6x + 3.5x2 = 0 Begin completing the square. Divide all terms by 3.5 the coefficient of the squared term: Divide each side by '3.5'. -21.71428571 + 1.714285714x + x2 = 0 Move the constant term to the right: Add '21.71428571' to each side of the equation. -21.71428571 + 1.714285714x + 21.71428571 + x2 = 0 + 21.71428571 Reorder the terms: -21.71428571 + 21.71428571 + 1.714285714x + x2 = 0 + 21.71428571 Combine like terms: -21.71428571 + 21.71428571 = 0.00000000 0.00000000 + 1.714285714x + x2 = 0 + 21.71428571 1.714285714x + x2 = 0 + 21.71428571 Combine like terms: 0 + 21.71428571 = 21.71428571 1.714285714x + x2 = 21.71428571 The x term is 1.714285714x. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714x + 0.7346938773 + x2 = 21.71428571 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714x + x2 = 21.71428571 + 0.7346938773 Combine like terms: 21.71428571 + 0.7346938773 = 22.4489795873 0.7346938773 + 1.714285714x + x2 = 22.4489795873 Factor a perfect square on the left side: (x + 0.857142857)(x + 0.857142857) = 22.4489795873 Calculate the square root of the right side: 4.738035414 Break this problem into two subproblems by setting (x + 0.857142857) equal to 4.738035414 and -4.738035414.Subproblem 1
x + 0.857142857 = 4.738035414 Simplifying x + 0.857142857 = 4.738035414 Reorder the terms: 0.857142857 + x = 4.738035414 Solving 0.857142857 + x = 4.738035414 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = 4.738035414 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = 4.738035414 + -0.857142857 x = 4.738035414 + -0.857142857 Combine like terms: 4.738035414 + -0.857142857 = 3.880892557 x = 3.880892557 Simplifying x = 3.880892557Subproblem 2
x + 0.857142857 = -4.738035414 Simplifying x + 0.857142857 = -4.738035414 Reorder the terms: 0.857142857 + x = -4.738035414 Solving 0.857142857 + x = -4.738035414 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = -4.738035414 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = -4.738035414 + -0.857142857 x = -4.738035414 + -0.857142857 Combine like terms: -4.738035414 + -0.857142857 = -5.595178271 x = -5.595178271 Simplifying x = -5.595178271Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.880892557, -5.595178271}
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