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Simplifying 7x(3x + -5) + (8x + 5) = 180 Reorder the terms: 7x(-5 + 3x) + (8x + 5) = 180 (-5 * 7x + 3x * 7x) + (8x + 5) = 180 (-35x + 21x2) + (8x + 5) = 180 Reorder the terms: -35x + 21x2 + (5 + 8x) = 180 Remove parenthesis around (5 + 8x) -35x + 21x2 + 5 + 8x = 180 Reorder the terms: 5 + -35x + 8x + 21x2 = 180 Combine like terms: -35x + 8x = -27x 5 + -27x + 21x2 = 180 Solving 5 + -27x + 21x2 = 180 Solving for variable 'x'. Reorder the terms: 5 + -180 + -27x + 21x2 = 180 + -180 Combine like terms: 5 + -180 = -175 -175 + -27x + 21x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -175 + -27x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -8.333333333 + -1.285714286x + x2 = 0 Move the constant term to the right: Add '8.333333333' to each side of the equation. -8.333333333 + -1.285714286x + 8.333333333 + x2 = 0 + 8.333333333 Reorder the terms: -8.333333333 + 8.333333333 + -1.285714286x + x2 = 0 + 8.333333333 Combine like terms: -8.333333333 + 8.333333333 = 0.000000000 0.000000000 + -1.285714286x + x2 = 0 + 8.333333333 -1.285714286x + x2 = 0 + 8.333333333 Combine like terms: 0 + 8.333333333 = 8.333333333 -1.285714286x + x2 = 8.333333333 The x term is -1.285714286x. Take half its coefficient (-0.642857143). Square it (0.4132653063) and add it to both sides. Add '0.4132653063' to each side of the equation. -1.285714286x + 0.4132653063 + x2 = 8.333333333 + 0.4132653063 Reorder the terms: 0.4132653063 + -1.285714286x + x2 = 8.333333333 + 0.4132653063 Combine like terms: 8.333333333 + 0.4132653063 = 8.7465986393 0.4132653063 + -1.285714286x + x2 = 8.7465986393 Factor a perfect square on the left side: (x + -0.642857143)(x + -0.642857143) = 8.7465986393 Calculate the square root of the right side: 2.957464901 Break this problem into two subproblems by setting (x + -0.642857143) equal to 2.957464901 and -2.957464901.Subproblem 1
x + -0.642857143 = 2.957464901 Simplifying x + -0.642857143 = 2.957464901 Reorder the terms: -0.642857143 + x = 2.957464901 Solving -0.642857143 + x = 2.957464901 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + x = 2.957464901 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + x = 2.957464901 + 0.642857143 x = 2.957464901 + 0.642857143 Combine like terms: 2.957464901 + 0.642857143 = 3.600322044 x = 3.600322044 Simplifying x = 3.600322044Subproblem 2
x + -0.642857143 = -2.957464901 Simplifying x + -0.642857143 = -2.957464901 Reorder the terms: -0.642857143 + x = -2.957464901 Solving -0.642857143 + x = -2.957464901 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.642857143' to each side of the equation. -0.642857143 + 0.642857143 + x = -2.957464901 + 0.642857143 Combine like terms: -0.642857143 + 0.642857143 = 0.000000000 0.000000000 + x = -2.957464901 + 0.642857143 x = -2.957464901 + 0.642857143 Combine like terms: -2.957464901 + 0.642857143 = -2.314607758 x = -2.314607758 Simplifying x = -2.314607758Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.600322044, -2.314607758}
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