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Simplifying 18x2 + -32x = 65 Reorder the terms: -32x + 18x2 = 65 Solving -32x + 18x2 = 65 Solving for variable 'x'. Reorder the terms: -65 + -32x + 18x2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + -32x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -3.611111111 + -1.777777778x + x2 = 0 Move the constant term to the right: Add '3.611111111' to each side of the equation. -3.611111111 + -1.777777778x + 3.611111111 + x2 = 0 + 3.611111111 Reorder the terms: -3.611111111 + 3.611111111 + -1.777777778x + x2 = 0 + 3.611111111 Combine like terms: -3.611111111 + 3.611111111 = 0.000000000 0.000000000 + -1.777777778x + x2 = 0 + 3.611111111 -1.777777778x + x2 = 0 + 3.611111111 Combine like terms: 0 + 3.611111111 = 3.611111111 -1.777777778x + x2 = 3.611111111 The x term is -1.777777778x. Take half its coefficient (-0.888888889). Square it (0.7901234570) and add it to both sides. Add '0.7901234570' to each side of the equation. -1.777777778x + 0.7901234570 + x2 = 3.611111111 + 0.7901234570 Reorder the terms: 0.7901234570 + -1.777777778x + x2 = 3.611111111 + 0.7901234570 Combine like terms: 3.611111111 + 0.7901234570 = 4.401234568 0.7901234570 + -1.777777778x + x2 = 4.401234568 Factor a perfect square on the left side: (x + -0.888888889)(x + -0.888888889) = 4.401234568 Calculate the square root of the right side: 2.097911954 Break this problem into two subproblems by setting (x + -0.888888889) equal to 2.097911954 and -2.097911954.Subproblem 1
x + -0.888888889 = 2.097911954 Simplifying x + -0.888888889 = 2.097911954 Reorder the terms: -0.888888889 + x = 2.097911954 Solving -0.888888889 + x = 2.097911954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.888888889' to each side of the equation. -0.888888889 + 0.888888889 + x = 2.097911954 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + x = 2.097911954 + 0.888888889 x = 2.097911954 + 0.888888889 Combine like terms: 2.097911954 + 0.888888889 = 2.986800843 x = 2.986800843 Simplifying x = 2.986800843Subproblem 2
x + -0.888888889 = -2.097911954 Simplifying x + -0.888888889 = -2.097911954 Reorder the terms: -0.888888889 + x = -2.097911954 Solving -0.888888889 + x = -2.097911954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.888888889' to each side of the equation. -0.888888889 + 0.888888889 + x = -2.097911954 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + x = -2.097911954 + 0.888888889 x = -2.097911954 + 0.888888889 Combine like terms: -2.097911954 + 0.888888889 = -1.209023065 x = -1.209023065 Simplifying x = -1.209023065Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.986800843, -1.209023065}
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