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Simplifying (2x + -3)(5x + -1) + (3x + -1)(x + -1) = 0 Reorder the terms: (-3 + 2x)(5x + -1) + (3x + -1)(x + -1) = 0 Reorder the terms: (-3 + 2x)(-1 + 5x) + (3x + -1)(x + -1) = 0 Multiply (-3 + 2x) * (-1 + 5x) (-3(-1 + 5x) + 2x * (-1 + 5x)) + (3x + -1)(x + -1) = 0 ((-1 * -3 + 5x * -3) + 2x * (-1 + 5x)) + (3x + -1)(x + -1) = 0 ((3 + -15x) + 2x * (-1 + 5x)) + (3x + -1)(x + -1) = 0 (3 + -15x + (-1 * 2x + 5x * 2x)) + (3x + -1)(x + -1) = 0 (3 + -15x + (-2x + 10x2)) + (3x + -1)(x + -1) = 0 Combine like terms: -15x + -2x = -17x (3 + -17x + 10x2) + (3x + -1)(x + -1) = 0 Reorder the terms: 3 + -17x + 10x2 + (-1 + 3x)(x + -1) = 0 Reorder the terms: 3 + -17x + 10x2 + (-1 + 3x)(-1 + x) = 0 Multiply (-1 + 3x) * (-1 + x) 3 + -17x + 10x2 + (-1(-1 + x) + 3x * (-1 + x)) = 0 3 + -17x + 10x2 + ((-1 * -1 + x * -1) + 3x * (-1 + x)) = 0 3 + -17x + 10x2 + ((1 + -1x) + 3x * (-1 + x)) = 0 3 + -17x + 10x2 + (1 + -1x + (-1 * 3x + x * 3x)) = 0 3 + -17x + 10x2 + (1 + -1x + (-3x + 3x2)) = 0 Combine like terms: -1x + -3x = -4x 3 + -17x + 10x2 + (1 + -4x + 3x2) = 0 Reorder the terms: 3 + 1 + -17x + -4x + 10x2 + 3x2 = 0 Combine like terms: 3 + 1 = 4 4 + -17x + -4x + 10x2 + 3x2 = 0 Combine like terms: -17x + -4x = -21x 4 + -21x + 10x2 + 3x2 = 0 Combine like terms: 10x2 + 3x2 = 13x2 4 + -21x + 13x2 = 0 Solving 4 + -21x + 13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. 0.3076923077 + -1.615384615x + x2 = 0 Move the constant term to the right: Add '-0.3076923077' to each side of the equation. 0.3076923077 + -1.615384615x + -0.3076923077 + x2 = 0 + -0.3076923077 Reorder the terms: 0.3076923077 + -0.3076923077 + -1.615384615x + x2 = 0 + -0.3076923077 Combine like terms: 0.3076923077 + -0.3076923077 = 0.0000000000 0.0000000000 + -1.615384615x + x2 = 0 + -0.3076923077 -1.615384615x + x2 = 0 + -0.3076923077 Combine like terms: 0 + -0.3076923077 = -0.3076923077 -1.615384615x + x2 = -0.3076923077 The x term is -1.615384615x. Take half its coefficient (-0.8076923075). Square it (0.6523668636) and add it to both sides. Add '0.6523668636' to each side of the equation. -1.615384615x + 0.6523668636 + x2 = -0.3076923077 + 0.6523668636 Reorder the terms: 0.6523668636 + -1.615384615x + x2 = -0.3076923077 + 0.6523668636 Combine like terms: -0.3076923077 + 0.6523668636 = 0.3446745559 0.6523668636 + -1.615384615x + x2 = 0.3446745559 Factor a perfect square on the left side: (x + -0.8076923075)(x + -0.8076923075) = 0.3446745559 Calculate the square root of the right side: 0.587089904 Break this problem into two subproblems by setting (x + -0.8076923075) equal to 0.587089904 and -0.587089904.Subproblem 1
x + -0.8076923075 = 0.587089904 Simplifying x + -0.8076923075 = 0.587089904 Reorder the terms: -0.8076923075 + x = 0.587089904 Solving -0.8076923075 + x = 0.587089904 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8076923075' to each side of the equation. -0.8076923075 + 0.8076923075 + x = 0.587089904 + 0.8076923075 Combine like terms: -0.8076923075 + 0.8076923075 = 0.0000000000 0.0000000000 + x = 0.587089904 + 0.8076923075 x = 0.587089904 + 0.8076923075 Combine like terms: 0.587089904 + 0.8076923075 = 1.3947822115 x = 1.3947822115 Simplifying x = 1.3947822115Subproblem 2
x + -0.8076923075 = -0.587089904 Simplifying x + -0.8076923075 = -0.587089904 Reorder the terms: -0.8076923075 + x = -0.587089904 Solving -0.8076923075 + x = -0.587089904 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8076923075' to each side of the equation. -0.8076923075 + 0.8076923075 + x = -0.587089904 + 0.8076923075 Combine like terms: -0.8076923075 + 0.8076923075 = 0.0000000000 0.0000000000 + x = -0.587089904 + 0.8076923075 x = -0.587089904 + 0.8076923075 Combine like terms: -0.587089904 + 0.8076923075 = 0.2206024035 x = 0.2206024035 Simplifying x = 0.2206024035Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3947822115, 0.2206024035}
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