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Simplifying x(3x + 5) = 105 Reorder the terms: x(5 + 3x) = 105 (5 * x + 3x * x) = 105 (5x + 3x2) = 105 Solving 5x + 3x2 = 105 Solving for variable 'x'. Reorder the terms: -105 + 5x + 3x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -105 + 5x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -35 + 1.666666667x + x2 = 0 Move the constant term to the right: Add '35' to each side of the equation. -35 + 1.666666667x + 35 + x2 = 0 + 35 Reorder the terms: -35 + 35 + 1.666666667x + x2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 1.666666667x + x2 = 0 + 35 1.666666667x + x2 = 0 + 35 Combine like terms: 0 + 35 = 35 1.666666667x + x2 = 35 The x term is 1.666666667x. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667x + 0.6944444447 + x2 = 35 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667x + x2 = 35 + 0.6944444447 Combine like terms: 35 + 0.6944444447 = 35.6944444447 0.6944444447 + 1.666666667x + x2 = 35.6944444447 Factor a perfect square on the left side: (x + 0.8333333335)(x + 0.8333333335) = 35.6944444447 Calculate the square root of the right side: 5.974482776 Break this problem into two subproblems by setting (x + 0.8333333335) equal to 5.974482776 and -5.974482776.Subproblem 1
x + 0.8333333335 = 5.974482776 Simplifying x + 0.8333333335 = 5.974482776 Reorder the terms: 0.8333333335 + x = 5.974482776 Solving 0.8333333335 + x = 5.974482776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = 5.974482776 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 5.974482776 + -0.8333333335 x = 5.974482776 + -0.8333333335 Combine like terms: 5.974482776 + -0.8333333335 = 5.1411494425 x = 5.1411494425 Simplifying x = 5.1411494425Subproblem 2
x + 0.8333333335 = -5.974482776 Simplifying x + 0.8333333335 = -5.974482776 Reorder the terms: 0.8333333335 + x = -5.974482776 Solving 0.8333333335 + x = -5.974482776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + x = -5.974482776 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -5.974482776 + -0.8333333335 x = -5.974482776 + -0.8333333335 Combine like terms: -5.974482776 + -0.8333333335 = -6.8078161095 x = -6.8078161095 Simplifying x = -6.8078161095Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.1411494425, -6.8078161095}
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