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Simplifying 90 = (-3x + 20)(-2x + -55) Reorder the terms: 90 = (20 + -3x)(-2x + -55) Reorder the terms: 90 = (20 + -3x)(-55 + -2x) Multiply (20 + -3x) * (-55 + -2x) 90 = (20(-55 + -2x) + -3x * (-55 + -2x)) 90 = ((-55 * 20 + -2x * 20) + -3x * (-55 + -2x)) 90 = ((-1100 + -40x) + -3x * (-55 + -2x)) 90 = (-1100 + -40x + (-55 * -3x + -2x * -3x)) 90 = (-1100 + -40x + (165x + 6x2)) Combine like terms: -40x + 165x = 125x 90 = (-1100 + 125x + 6x2) Solving 90 = -1100 + 125x + 6x2 Solving for variable 'x'. Combine like terms: 90 + 1100 = 1190 1190 + -125x + -6x2 = -1100 + 125x + 6x2 + 1100 + -125x + -6x2 Reorder the terms: 1190 + -125x + -6x2 = -1100 + 1100 + 125x + -125x + 6x2 + -6x2 Combine like terms: -1100 + 1100 = 0 1190 + -125x + -6x2 = 0 + 125x + -125x + 6x2 + -6x2 1190 + -125x + -6x2 = 125x + -125x + 6x2 + -6x2 Combine like terms: 125x + -125x = 0 1190 + -125x + -6x2 = 0 + 6x2 + -6x2 1190 + -125x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 1190 + -125x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -198.3333333 + 20.83333333x + x2 = 0 Move the constant term to the right: Add '198.3333333' to each side of the equation. -198.3333333 + 20.83333333x + 198.3333333 + x2 = 0 + 198.3333333 Reorder the terms: -198.3333333 + 198.3333333 + 20.83333333x + x2 = 0 + 198.3333333 Combine like terms: -198.3333333 + 198.3333333 = 0.0000000 0.0000000 + 20.83333333x + x2 = 0 + 198.3333333 20.83333333x + x2 = 0 + 198.3333333 Combine like terms: 0 + 198.3333333 = 198.3333333 20.83333333x + x2 = 198.3333333 The x term is 20.83333333x. Take half its coefficient (10.41666667). Square it (108.5069445) and add it to both sides. Add '108.5069445' to each side of the equation. 20.83333333x + 108.5069445 + x2 = 198.3333333 + 108.5069445 Reorder the terms: 108.5069445 + 20.83333333x + x2 = 198.3333333 + 108.5069445 Combine like terms: 198.3333333 + 108.5069445 = 306.8402778 108.5069445 + 20.83333333x + x2 = 306.8402778 Factor a perfect square on the left side: (x + 10.41666667)(x + 10.41666667) = 306.8402778 Calculate the square root of the right side: 17.516856961 Break this problem into two subproblems by setting (x + 10.41666667) equal to 17.516856961 and -17.516856961.Subproblem 1
x + 10.41666667 = 17.516856961 Simplifying x + 10.41666667 = 17.516856961 Reorder the terms: 10.41666667 + x = 17.516856961 Solving 10.41666667 + x = 17.516856961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.41666667' to each side of the equation. 10.41666667 + -10.41666667 + x = 17.516856961 + -10.41666667 Combine like terms: 10.41666667 + -10.41666667 = 0.00000000 0.00000000 + x = 17.516856961 + -10.41666667 x = 17.516856961 + -10.41666667 Combine like terms: 17.516856961 + -10.41666667 = 7.100190291 x = 7.100190291 Simplifying x = 7.100190291Subproblem 2
x + 10.41666667 = -17.516856961 Simplifying x + 10.41666667 = -17.516856961 Reorder the terms: 10.41666667 + x = -17.516856961 Solving 10.41666667 + x = -17.516856961 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.41666667' to each side of the equation. 10.41666667 + -10.41666667 + x = -17.516856961 + -10.41666667 Combine like terms: 10.41666667 + -10.41666667 = 0.00000000 0.00000000 + x = -17.516856961 + -10.41666667 x = -17.516856961 + -10.41666667 Combine like terms: -17.516856961 + -10.41666667 = -27.933523631 x = -27.933523631 Simplifying x = -27.933523631Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.100190291, -27.933523631}
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