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Simplifying 80x2 + -135x + 54 = 0 Reorder the terms: 54 + -135x + 80x2 = 0 Solving 54 + -135x + 80x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 80 the coefficient of the squared term: Divide each side by '80'. 0.675 + -1.6875x + x2 = 0 Move the constant term to the right: Add '-0.675' to each side of the equation. 0.675 + -1.6875x + -0.675 + x2 = 0 + -0.675 Reorder the terms: 0.675 + -0.675 + -1.6875x + x2 = 0 + -0.675 Combine like terms: 0.675 + -0.675 = 0.000 0.000 + -1.6875x + x2 = 0 + -0.675 -1.6875x + x2 = 0 + -0.675 Combine like terms: 0 + -0.675 = -0.675 -1.6875x + x2 = -0.675 The x term is -1.6875x. Take half its coefficient (-0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. -1.6875x + 0.7119140625 + x2 = -0.675 + 0.7119140625 Reorder the terms: 0.7119140625 + -1.6875x + x2 = -0.675 + 0.7119140625 Combine like terms: -0.675 + 0.7119140625 = 0.0369140625 0.7119140625 + -1.6875x + x2 = 0.0369140625 Factor a perfect square on the left side: (x + -0.84375)(x + -0.84375) = 0.0369140625 Calculate the square root of the right side: 0.192130327 Break this problem into two subproblems by setting (x + -0.84375) equal to 0.192130327 and -0.192130327.Subproblem 1
x + -0.84375 = 0.192130327 Simplifying x + -0.84375 = 0.192130327 Reorder the terms: -0.84375 + x = 0.192130327 Solving -0.84375 + x = 0.192130327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = 0.192130327 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = 0.192130327 + 0.84375 x = 0.192130327 + 0.84375 Combine like terms: 0.192130327 + 0.84375 = 1.035880327 x = 1.035880327 Simplifying x = 1.035880327Subproblem 2
x + -0.84375 = -0.192130327 Simplifying x + -0.84375 = -0.192130327 Reorder the terms: -0.84375 + x = -0.192130327 Solving -0.84375 + x = -0.192130327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = -0.192130327 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = -0.192130327 + 0.84375 x = -0.192130327 + 0.84375 Combine like terms: -0.192130327 + 0.84375 = 0.651619673 x = 0.651619673 Simplifying x = 0.651619673Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.035880327, 0.651619673}
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