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Simplifying 10(x + -0.2) * 2 + -100(0.3x + 5)(0.3x + -5) + -1(x + -50) * 2 = 0 Reorder the terms: 10(-0.2 + x) * 2 + -100(0.3x + 5)(0.3x + -5) + -1(x + -50) * 2 = 0 Reorder the terms for easier multiplication: 10 * 2(-0.2 + x) + -100(0.3x + 5)(0.3x + -5) + -1(x + -50) * 2 = 0 Multiply 10 * 2 20(-0.2 + x) + -100(0.3x + 5)(0.3x + -5) + -1(x + -50) * 2 = 0 (-0.2 * 20 + x * 20) + -100(0.3x + 5)(0.3x + -5) + -1(x + -50) * 2 = 0 (-4 + 20x) + -100(0.3x + 5)(0.3x + -5) + -1(x + -50) * 2 = 0 Reorder the terms: -4 + 20x + -100(5 + 0.3x)(0.3x + -5) + -1(x + -50) * 2 = 0 Reorder the terms: -4 + 20x + -100(5 + 0.3x)(-5 + 0.3x) + -1(x + -50) * 2 = 0 Multiply (5 + 0.3x) * (-5 + 0.3x) -4 + 20x + -100(5(-5 + 0.3x) + 0.3x * (-5 + 0.3x)) + -1(x + -50) * 2 = 0 -4 + 20x + -100((-5 * 5 + 0.3x * 5) + 0.3x * (-5 + 0.3x)) + -1(x + -50) * 2 = 0 -4 + 20x + -100((-25 + 1.5x) + 0.3x * (-5 + 0.3x)) + -1(x + -50) * 2 = 0 -4 + 20x + -100(-25 + 1.5x + (-5 * 0.3x + 0.3x * 0.3x)) + -1(x + -50) * 2 = 0 -4 + 20x + -100(-25 + 1.5x + (-1.5x + 0.09x2)) + -1(x + -50) * 2 = 0 Combine like terms: 1.5x + -1.5x = 0.0 -4 + 20x + -100(-25 + 0.0 + 0.09x2) + -1(x + -50) * 2 = 0 -4 + 20x + -100(-25 + 0.09x2) + -1(x + -50) * 2 = 0 -4 + 20x + (-25 * -100 + 0.09x2 * -100) + -1(x + -50) * 2 = 0 -4 + 20x + (2500 + -9x2) + -1(x + -50) * 2 = 0 Reorder the terms: -4 + 20x + 2500 + -9x2 + -1(-50 + x) * 2 = 0 Reorder the terms for easier multiplication: -4 + 20x + 2500 + -9x2 + -1 * 2(-50 + x) = 0 Multiply -1 * 2 -4 + 20x + 2500 + -9x2 + -2(-50 + x) = 0 -4 + 20x + 2500 + -9x2 + (-50 * -2 + x * -2) = 0 -4 + 20x + 2500 + -9x2 + (100 + -2x) = 0 Reorder the terms: -4 + 2500 + 100 + 20x + -2x + -9x2 = 0 Combine like terms: -4 + 2500 = 2496 2496 + 100 + 20x + -2x + -9x2 = 0 Combine like terms: 2496 + 100 = 2596 2596 + 20x + -2x + -9x2 = 0 Combine like terms: 20x + -2x = 18x 2596 + 18x + -9x2 = 0 Solving 2596 + 18x + -9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -288.4444444 + -2x + x2 = 0 Move the constant term to the right: Add '288.4444444' to each side of the equation. -288.4444444 + -2x + 288.4444444 + x2 = 0 + 288.4444444 Reorder the terms: -288.4444444 + 288.4444444 + -2x + x2 = 0 + 288.4444444 Combine like terms: -288.4444444 + 288.4444444 = 0.0000000 0.0000000 + -2x + x2 = 0 + 288.4444444 -2x + x2 = 0 + 288.4444444 Combine like terms: 0 + 288.4444444 = 288.4444444 -2x + x2 = 288.4444444 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = 288.4444444 + 1 Reorder the terms: 1 + -2x + x2 = 288.4444444 + 1 Combine like terms: 288.4444444 + 1 = 289.4444444 1 + -2x + x2 = 289.4444444 Factor a perfect square on the left side: (x + -1)(x + -1) = 289.4444444 Calculate the square root of the right side: 17.013066872 Break this problem into two subproblems by setting (x + -1) equal to 17.013066872 and -17.013066872.Subproblem 1
x + -1 = 17.013066872 Simplifying x + -1 = 17.013066872 Reorder the terms: -1 + x = 17.013066872 Solving -1 + x = 17.013066872 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 17.013066872 + 1 Combine like terms: -1 + 1 = 0 0 + x = 17.013066872 + 1 x = 17.013066872 + 1 Combine like terms: 17.013066872 + 1 = 18.013066872 x = 18.013066872 Simplifying x = 18.013066872Subproblem 2
x + -1 = -17.013066872 Simplifying x + -1 = -17.013066872 Reorder the terms: -1 + x = -17.013066872 Solving -1 + x = -17.013066872 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -17.013066872 + 1 Combine like terms: -1 + 1 = 0 0 + x = -17.013066872 + 1 x = -17.013066872 + 1 Combine like terms: -17.013066872 + 1 = -16.013066872 x = -16.013066872 Simplifying x = -16.013066872Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.013066872, -16.013066872}
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