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Simplifying 4x + 7x(x + -3) = 67 Reorder the terms: 4x + 7x(-3 + x) = 67 4x + (-3 * 7x + x * 7x) = 67 4x + (-21x + 7x2) = 67 Combine like terms: 4x + -21x = -17x -17x + 7x2 = 67 Solving -17x + 7x2 = 67 Solving for variable 'x'. Reorder the terms: -67 + -17x + 7x2 = 67 + -67 Combine like terms: 67 + -67 = 0 -67 + -17x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -9.571428571 + -2.428571429x + x2 = 0 Move the constant term to the right: Add '9.571428571' to each side of the equation. -9.571428571 + -2.428571429x + 9.571428571 + x2 = 0 + 9.571428571 Reorder the terms: -9.571428571 + 9.571428571 + -2.428571429x + x2 = 0 + 9.571428571 Combine like terms: -9.571428571 + 9.571428571 = 0.000000000 0.000000000 + -2.428571429x + x2 = 0 + 9.571428571 -2.428571429x + x2 = 0 + 9.571428571 Combine like terms: 0 + 9.571428571 = 9.571428571 -2.428571429x + x2 = 9.571428571 The x term is -2.428571429x. Take half its coefficient (-1.214285715). Square it (1.474489798) and add it to both sides. Add '1.474489798' to each side of the equation. -2.428571429x + 1.474489798 + x2 = 9.571428571 + 1.474489798 Reorder the terms: 1.474489798 + -2.428571429x + x2 = 9.571428571 + 1.474489798 Combine like terms: 9.571428571 + 1.474489798 = 11.045918369 1.474489798 + -2.428571429x + x2 = 11.045918369 Factor a perfect square on the left side: (x + -1.214285715)(x + -1.214285715) = 11.045918369 Calculate the square root of the right side: 3.323540036 Break this problem into two subproblems by setting (x + -1.214285715) equal to 3.323540036 and -3.323540036.Subproblem 1
x + -1.214285715 = 3.323540036 Simplifying x + -1.214285715 = 3.323540036 Reorder the terms: -1.214285715 + x = 3.323540036 Solving -1.214285715 + x = 3.323540036 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + x = 3.323540036 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + x = 3.323540036 + 1.214285715 x = 3.323540036 + 1.214285715 Combine like terms: 3.323540036 + 1.214285715 = 4.537825751 x = 4.537825751 Simplifying x = 4.537825751Subproblem 2
x + -1.214285715 = -3.323540036 Simplifying x + -1.214285715 = -3.323540036 Reorder the terms: -1.214285715 + x = -3.323540036 Solving -1.214285715 + x = -3.323540036 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + x = -3.323540036 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + x = -3.323540036 + 1.214285715 x = -3.323540036 + 1.214285715 Combine like terms: -3.323540036 + 1.214285715 = -2.109254321 x = -2.109254321 Simplifying x = -2.109254321Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.537825751, -2.109254321}
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